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PROPERTIES OF MATTER

BY THE SAME AUTHOR.

LIGHT.

Third Edition. In crown 8vo, cloth, price 7s. 6cL

DYNAMICS.

In crown 8vo, cloth, price 7s. 6d.

NEWTON'S LAWS OF MOTION.

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PROPERTIES OF MATTER

P. G. TAIT, M.A., SEC. R.S.E.

HONORARY FELLOW OF ST. PKTKR'S COLLEGE, CAMBRIDGE,

PROFESSOR OP NATURAL PHILOSOPHY IN THE

UNIVERSITY OF EDINBURGH

FIFTH EDITION

EDITED BY

W. PEDDIE, D.Sc., F.R.S.E.

HARRIS PROFESSOR OF PHYSIOS IN UNIVERSITY COLLEGE, DUNDER,
UNIVERSITY OF ST. ANDREWS

LONDON
ADAM AND CHARLES BLACK

1907

T3

First Edition published April, 1885 ;

Second Edition, October, 1890 ; Third Edition, June, 1894 ;
Fourth Edition, November, 1899 ; Fifth Edition, July, 1907.

The Right of Translation and Reproduction is Reserved

PREFACE.

IT is desirable that a work so characteristic as this is
should remain, in a new edition, as far as possible in the
form in which it left the hands of its distinguished
author. The rapid advance of physical science, in the
few years which have passed since the last edition
appeared, has necessitated some slight additions. In
making these, the original plan of the book has been
strictly adhered to, and all additions have been placed
within brackets and initialed.

The work of revision has been a pleasure to me, as a
service gladly rendered to the memory of my former
master and friend.

W. PEDDIE.

UNIVERSITY COLLEGE, DUNDEE,
June 3, 1907.

222568

PREFACE TO THE FIRST EDITION.

THE subject of this elementary work still forms in
accordance with tradition from the days of Eobison,
Playfair, Leslie, and Forbes the introduction to the
course of Natural Philosophy in Edinburgh University.

The work is (with the exception of a few isolated
sections) intended for the average student ; who is sup-
posed to have a sound knowledge of ordinary Geometry,
and a moderate acquaintance with the elements of
Algebra and of Trigonometry.

But he is also supposed to have what he can easily
obtain from the simpler parts of the two first chapters of
Thomson and Tait's Elements of Natural Philosophy, or
from Clerk-Maxwell's excellent little treatise on Matter
and Motion a general acquaintance with the fundamental
principles of Kinematics of a Point and of Kinetics of
a Particle. To have treated these subjects at greater
length than has here been attempted would have rendered
it imperative to omit much of the development of im-
portant parts of preliminary Physics, of which, so far as
I know, there is no modern British text-book. The
work was peremptorily limited to a small volume; so
that the parts of these auxiliary subjects which have

viii PREFACE TO THE FIRST EDITION.

been admitted are mainly of two kinds : those which
are really introductory to the books just mentioned,
because treating of matters' usually deemed too simple
for special notice ; and a few which are in a sense sup-
plementary, because giving valuable results not usually
included in elementary books.

It is my present intention to complete my series of
text-books by similar volumes on Dynamics, Sound, and
Electricity. Should I succeed in bringing out such
works, I shall thenceforth be enabled to introduce
references to one or other, instead of the digressions
which are absolutely necessary in every self-contained
elementary treatise devoted to one special branch of
Physics only.

P. G. TAIT.

COLLEGE, EDINBURGH,
March 5, 1885.

PREFACE TO THE SECOND EDITION.

LN the present edition this Treatise has been carefully
revised and considerably extended : special attention
having been paid to passages where a difficulty had
been found.

For one of the most important additions I am indebted
to M. Amagat, who has very kindly enabled me to avail
myself of some of his splendid but hitherto unpublished
results. These relate to the compression of fluids exposed
to enormous pressures ; and, when published entire, will
form a singularly interesting and practically new branch
jf the subject.

To some of the scientific critics of the first edition I
am indebted for suggestions of real value, and I have
endeavoured to profit by them. I must except, however,
those which concern my treatment of the subject of Force.
I have seen so much mischief done by this quasi-personi-
fication of a mere sense -impression that, even in an
elementary book, I am constrained to protest against it.
(See 15 of the text.) I feel assured that the difficulties
which are now everywhere felt as to the great scientific
question of the day, the nature of what we call electricity,
ire in great part due to the way in which our modes of

x PREFACE TO THE SECOND EDITION.

thinking have been, by early training and subsequent
habit, encouraged to run in this fatal groove.

To some of my other critics, more aggressive because
less scientific, I have been indebted for genuine amuse-
ment. Nothing is, however, without its use in this
world, though it may occasionally be difficult to discover
that use. It would seem, then, that the function of the
unscientific critics of a scientific book is (like that of the
writers of slipshod English) to furnish examiners with
rich material for questions of the well-known kind :
"Point out all the errors in the following passage."
Nothing is more difficult than the attempt to make such
passages : and the results are usually forced and awkward.
From the critics I allude to they come in perfection.

There is one additional remark which I must make.
The majority of the illustrations in this work (whether
given in words or by diagrams) are, when the contrary is
not stated, to the best of my knowledge original. I make
the remark lest I should be supposed to have taken them
from some of the books in which they have been re-
produced without acknowledgment of their source. It is
flattering to have one's work thus appreciated, but the
honour has its little inconveniences.

P. G. TAIT.

COLLEGE, EDINBURGH,
July 1, 1890.

PREFACE TO THE THIRD EDITION.

IN spite of what I must still consider a tolerably com-
plete statement of my position ( 11-15, 108, etc.), some
of my critics persist in accusing me of deliberate incon-
sistency in my treatment of Force. I am represented by
them as first telling my readers that there is "no such
thing as Force," and then introducing utter confusion by
the assertion that "Matter is merely the plaything of
Force" ! They appear to be unable to perceive that the
idea of Force is an essential feature of Newton's Laws of
Motion : and that, until we are provided with an effi-
cient substitute for Newton's system, we must retain it
in its integrity. To have left these statements altogether
unnoticed might have been prejudicial to the book itself,
in the eyes of the many weak ones who regard the dicta of
a critic (especially when he is anonymous) as necessarily
authoritative. To take more than this passing notice of
them would be to exaggerate their importance.

Since the last edition of this book was published, the
whole of M. Amagat's splendid experimental results have
become generally accessible : and I have made consider-
able additional use of them, especially in Chaps. IX.
andX.

Parts of Chap. VIII. have been considerably modified,

xiv PREFACE.

and VI. of this work may be referred to a little pam-
phlet on Newton' 's Laws of Motion, which has just been
issued by the same publishers. In it I have discussed,
though with studied conciseness, a good deal of important
matter which the plan of this volume prevented me from
introducing.

The present issue has been kept up to date, so far as
recent real extensions of the subjects it treats of are con-
cerned. Other engrossing work has (happily ?) rendered
impracticable for the moment a scheme which I had con-
templated for a thorough revision of the whole plan of the
volume.

P. G. TAIT.

COLLEGE, EDINBURGH,
September 19, 1899.

CONTENTS.

CHAPTER I.

PAG*

INTRODUCTORY . . . . . ' . 1

CHAPTER II.

SOME HYPOTHESES AS TO THE ULTIMATE STRUCTURE OF

MATTER ...... 18

CHAPTER III.

EXAMPLES OF TERMS IN COMMON USE AS APPLIED TO

MATTER ...... 25

CHAPTER IV.
TIME AND SPACE . . . ... 48

CHAPTER V.
IMPENETRABILITY, POROSITY, DIVISIBILITY . . 83

CHAPTER VI.
INERTIA, MOBILITY, CENTRIFUGAL FORCE . . 94

CHAPTER VII.

GRAVITATION . . . . . .113

CHAPTER VIII.
PRELIMINARY TO DEFORMABILITY AND ELASTICITY . 146

xvi CONTENTS.

CHAPTER IX.

PAGE

COMPRESSIBILITY OF GASES AND VAPOURS . . 161

CHAPTER X.

COMPRESSION OF LIQUIDS . . . . .188

CHAPTER XI.
COMPRESSIBILITY AND RIGIDITY OF SOLIDS . . 205

CHAPTER XII.
COHESION AND CAPILLARITY .... 242

CHAPTER XIII.

DIFFUSION, OSMOSE, TRANSPIRATION, VISCOSITY, ETC. . 275

CHAPTER XIV.

AGGREGATION OF PARTICLES . . . 296

CHAPTER XV.
DISINTEGRATION OF THE ATOM . . . .311

APPENDIX.
I. HYPOTHESES AS TO THE CONSTITUTION OF MATTER.

BY PROFESSOR FLINT, D.D. . . .316

II. EXTRACTS FROM CLERK-MAXWELL'S ARTICLE "ATOM" 320

III. VITRUVIUS ON ARCHIMEDES' EXPERIMENT . . 335

IV* NOTE ON A SINGULAR PASSAGE IN THE " PRINCIPIA " 336

V. EXTRACT FROM LA IIMOR'S "AETHER AND MATTER" 344

VI. EXTRACT FROM KELVIN'S "BALTIMORE LECTURES" 346

INDEX 350

PKOPEKTIES OF MATTEE.

CHAPTER I.

INTRODUCTORY.

1. WE start with certain assumptions or AXIOMS, which
are by no means of an a priori character, having been
forced upon us by the observations and experiments of
many generations :

(1) The physical universe has an objective existence.

(2) We become cognisant of it solely by the aid of our

Senses.

(3) The indications of the Senses are always imperfect,

and often misleading ; but

(4) The patient exercise of Reason enables us to control

these indications, and gradually, but surely, to
sift truth from falsehood.

2. If, for a moment, we use the word Thing to denote,
generally, whatever we are constrained to allow has
objective existence : i.e. exists altogether independently
of our senses and of our reason : we arrive at the follow-
ing conclusions :

A. In the physical universe there are but two classes
of things, MATTER and ENERGY.

A

2 ' PROPERTY "o

B. TIME and SPACE, though well known to all (in
Newton's words, omnibus not!ssima\ are not things. 1

C. NUMBER, MAGNITUDE, POSITION, VELOCITY, etc., are
likewise not things.

D. CONSCIOUSNESS, VOLITION, eetu, are not physical.

3. So says modern physical science, and to its generally
received statements we cannot but adhere.

Metaphysicians, of course, who trust entirely to so-
called " light of nature," have their own views on this,
as on all other subjects ; but the number and variety of
these views, some of which are entirely incompatible
with others, form a striking contrast to the general con-
sensus of opinion on the part of those who have at least
tried to deserve to know.

In the words of v. Helmholtz, 2 one of the chief modern
authorities in science properly so-called :

"The genuine metaphysician, in view of a presumed
necessity of thought, looks down with an air of superiority
on those who labour to investigate the facts. Has it
already been forgotten how much mischief this procedure

1 " Space is . . . regarded as a condition of the possibility of
phenpmena, not as a determination produced by them ; it is a
representation d. priori which necessarily precedes all external
phenomena : "

"Time is not an empirical concept deduced from any experience,
for neither) co-existence nor succession would enter into our per-
ception, if the representation of time were not given d priori." -
KANT, Critique of Pure Reason ; Max Miiller's Translation.

2 " Hier haben wir den achten Metaphysiker. Einer angeblichen
Denknothwendigkeit gegeniiber blickt er hochmiithig auf die,
welche sich um Erforschung der Thatsachen bemiihen, herab. 1st
es schon vergessen, wie viel Unheil dieses Verfahren in den
friiheren Entwicklungsperioden der Naturwissenschaften ange-
richtet hat ? " Preface to the German Translation of the second
part of Thomson and Tail's Natural Philosophy.

INTRODUCTORY. 3

wrought in the earlier stages of the development of the
sciences 1 "

Clerk-Maxwell develops the contrast somewhat more
elaborately :

"... In every human pursuit there are two courses
one, that which in its lowest form is called the useful, and
has for its ultimate object the extension of knowledge,
the dominion over Nature, and the welfare of mankind
The objects of the second course are entirely self-con-
tained. Theories are elaborated for theories' sake, diffi-
culties are sought out and treasured as such, and no
argument is to be considered perfect unless it lands the
reasoner at the point from which he started. . . .

" The education of man is so well provided for in the
world around him, and so hopeless in any of the worlds
which he makes for himself, that it becomes of the
utmost importance to distinguish natural truth from
artificial system, the development of a science from the
envelopment of a craft."

Newton, however, had long before expressed essentially
the same ideas. He said :

"To tell us that every species of things is endowed
with an occult specific quality, by which it acts and pro-
duces manifest effects, is to tell us nothing; but to
derive two or three general principles of motion from
phenomena, and afterwards to tell us how the proper-
ties of all corporeal things follow from those manifest
principles, would be a very great step in philosophy,
though the causes of those principles were not yet dis-
covered ; and therefore I scruple not to propose the
principles of motion above mentioned, they being of
very general extent, and leave the causes to be found
out."

4 PROPERTIES OF MATTER.

Midway between Newton's time and our own, another
very great man, Young, spoke as follows of the pernicious
effects of metaphysics in the ancient world :

" A one of the departments of human knowledge were
excluded from the pursuits ... of the Grecian sages,
until Socrates introduced, into the Ionian school, a taste
for metaphysical speculations, which excluded almost all
disposition to reason coolly and clearly on natural causes
and effects."

Quotations like these might be multiplied indefinitely.
But we have given enough to j ustify fully the statements
made in the opening section. These statements must be
our guide in all that follows.

4. A stone, a piece of lead or brass, water, air, the
ether or luminiferous medium, etc., are portions of
Matter; wound-up springs, water-power, wind, waves,
compressed air, hot bodies, electric currents, as well as the
objective phenomena corresponding to our sensations of
sound and light, are examples of Energy associated with
matter. For the present, at all events, we are in the
habit of speaking of Energy as Kinetic when it obviously,
or at least certainly, depends on motion of matter. Thus
the energy of wind, waves, heat, electric currents, etc., is
kinetic. But energy stored up, as in a charged thunder-
cloud, a wound-up spring, a "head" of water, is called
Potential. Of its intimate nature we as yet know
nothing, and it is possible that it may be ultimately
kinetic : i.e. dependent upon motions of inscrutably
minute parts of matter.

5. All trustworthy experiments, without exception,
have been found to lead to the conviction that matter is
unalterable in quantity by any process at the command
of man.

INTRODUCTORY. 5

This is one of the strongest arguments in favour of
the objective existence of matter. It was usefully
employed, at the very end of last century, by Rumford in
his memorable Inquiry concerning the Source of the Heat
excited by Friction. 1

It forms also the indispensable foundation of modern
chemistry, whose main instrument is the balance, used
to determine quantity of matter with great exactness.

We may speak of this property, for the sake of future
reference, as the Conservation of Matter. It justifies one-
half of the statement in 2, A.

It is to be remarked here that the statements just made,
being the direct result of experiment, are strictly applicable
to gross matter only. The Ether or luminiferous and
electrical medium is certainly matter, in the sense of
having Inertia ( 9), but we have at present no means of
investigating its conservation.

6. So far the reader (if he resemble at all the average
student of our acquaintance) is not likely to feel much
difficulty. His every-day experience must have long ago
impressed on him the conviction of the objectivity of
matter, though perhaps he may not have learned to express
it in such a form of words.

But it is usually otherwise when he is told that energy
has an objective existence quite as certainly as has matter.
He has been accustomed to the working of water-mills, let
us say, and he cannot but allow that a " head " of water is
something other than the water ; it is something associated
with the water in virtue of its elevation. He sees and (if
he be of an economic turn) he deplores the terrible waste
of water-power which is stupidly permitted to go on all
over the world. He allows that water-power does exist,
1 Phil. Trans., 1798.

6 PROPERTIES OF MATTER.

but the waste which he laments he looks upon as its
annihilation. Till within the last fifty years or so the
vast majority even of scientific men held precisely the
same opinion.

7. The modern doctrine of the Conservation of Energy,
securely based upon the splendid investigations of Joule
and others, completes the justification of our preliminary
statement. Energy, like matter, has been experimentally
proved to be indestructible and uncreatable by man. It
exists, therefore, altogether independently of human
senses and human reason, though it is known to man
solely by their aid.

One of the most curious passages in history is that
which describes the quest of The Perpetual Motion.
This was simply the attempt to discover a continuous
Source of fresh mechanical energy. In 1775 the Academy
of Sciences declined, for the future, to consider any
scheme which professed to furnish work without corre-
sponding and equivalent expenditure. But the race of
Perpetual Motionists is by no means even yet extinct.
The doctrine of the impossibility of the Perpetual Motion
is often valuable in modern physics (see, for instance,
139 below), as it furnishes simple ex absurdo proofs of
important fundamental theorems.

The objectivity of energy is virtually admitted in a
curious way, by its being advertised for sale. Thus in
manufacturing centres, where a mill-owner has a steam-
engine too powerful for his requirements, he issues a
notice to the effect, " Spare Power to let." But, of
course, the common phrase "price of labour" at once
acknowledges the objectivity of work.

8. There is, however,, a most important point to be
noticed. Energy is never found except in association

INTRODUCTORY. 7

with matter. Hence we might define matter as the
Vehicle or Receptacle of Energy ; and it is already more
than probable that energy will ultimately be found, in
all its varied forms, to depend upon Motion of matter.

This is advanced, for the moment, as a mere introductory
statement, instances of which will be discussed even in
the present work ; but its complete treatment would
require the introduction of branches of physics with
which we have here nothing to do. One great argument
in its favour is, that matter is found to consist of parts
which preserve their identity, while energy is manifested
to us only in the act of transformation, and (though
measurable) cannot be identified. For this is precisely
what we should expect to find if energy depends in-
variably on motion of matter.

9. Beside their common characteristic, conservation,
and in strange contrast to it, we have their characteristic
difference. Matter is simply passive (inert is the scientific
word) ; energy is perpetually undergoing transformation.
The one is, as it were, the body of the physical universe;
the other its life and activity. All terrestrial phenomena,
from winds and waves to lightning and thunder, eruptions
and earthquakes, are transformations of energy. So
are alike the brief flash of a falling star, and the fiery
glow from the mighty solar outbursts of incandescent
hydrogen.

10. From the strictly scientific point of view, the greater
part of the present work would be said to deal with energy
rather than with matter. In fact, were we to speak of
weight as a property of matter, in the sense that a stone
of itself has weight, or even in the sense that the earth
attracts the stone, we should go directly in the teeth of
Newton's distinct assertion.

8 PROPERTIES OF MATTER.

For such a statement (because confined to the attract-
ing bodies alone) implies the existence of Action at a
Distance, a very old but most pernicious heresy, of which
much more than traces still exist among certain schools,
even of physicists. (See Newton's words on this subject,
160 below.)

Gravitation, like all other mutual actions between
particles of matter, such as give rise to cohesion,
elasticity, etc., must, with our present knowledge, be set
down to the energy which particles of matter are
found to possess when separated. The intervening
mechanism by which this is to be accounted for has, as
yet, only been guessed at, and none of the guesses have
been succesoM. Clerk-Maxwell's success in explaining
electric and magnetic attractions by something analogous
to stresses and rotations in the luminiferous ether shows,
however, that we need not despair of being able to dis-
cover the ultimate mechanism of gravitation.

But there is great convenience in separating, as far
as possible, the treatment of Mass, Weight, Cohesion,
Elasticity, Viscosity, etc., which we range under the
general title, Properties of Matter, from that of Heat,
Light, Electric Energy, etc., which can all in great
measure be studied without express reference to any one
special kind of matter though, as forms of energy, they
exist only ( 8 above) in association with matter. Along
with these forms of energy must of course be treated the
allied properties of matter, such as specific heat, refractive
index, conductivity, etc. Such, therefore, are foreign to
the present work. And it must be remarked that, even
in popular language, we invariably speak of the hardness
of a body, its rigidity, its elasticity, as belonging to it in
much the same sense as does its density or its atomic

INTRODUCTORY. 9

weight and certainly in a much more intimate sense
than does its temperature or its electric potential.

It is, therefore, on the two grounds of custom and
convenience that we use the term Properties of Matter as
the title of this work. The error involved is not by any
means so monstrous as that which all agree to perpetuate
hy the use of the term Centrifugal Force.

11. The word Force must often, were it only for
brevity's sake, be used in the present work. As it does
not denote either matter or energy, it is not a term for
anything objective ( 2, A). The idea it is meant to
express is suggested to us by the " muscular sense," just
as the ideas of brightness, noise, smell, or pain are sug-
gested by other senses : though they do not correspond
directly to anything which exists outside us.

It is exceedingly difficult to realize fully the fact that
noise is a mere subjective impression, even when reason
has convinced us that outside the drum of the ear then-
is nothing to correspond to it except a periodic com
pression and dilatation of the air.

Still more difficult is it to realize that outside us all it
dark; and that the objective cause of even the most
gorgeous of optical phenomena is an excessively rapid
quivering motion of the ethereal jelly which extends-
through all space.

We need not, therefore, be surprised at the tenacity
with which the great majority, even of scientific men,
still cling to the notion of force as something objective.

But if it were objective, what an absolutely astounding
difficulty would have to be faced by one who tries to
explain the nature of hydrostatic pressure; and who
finds that by the touch of a finger on a little piston he
can produce a pressure of (say) a pound weight on every

10 PROPERTIES OF MATTER.

square inch of the surface of a vessel, however large, if
full of water, and the same amount on every square inch
of surface of every object immersed in it, even if that
object consisted of hundreds of square miles of sheets
of tinfoil far enough apart to let the water penetrate
between them. All this, moreover, is found to disappear
the moment he lifts his finger !

When we communicate energy to a body, as in pushing
or drawing a carriage, the impression produced upon our
muscular sense does not correspond to the energy com-
municated per second, but to the energy communicated
per inch of the motion. For experiment has proved
that what appears to our muscular sense as a definite
tension (in a cord, let us say) is associated with the com-
munication of energy, to any mass of matter whatever,
in direct proportion to the (linear) space through which
it is exerted, altogether independently of the speed with
which the mass may be already moving in the direction
of the tension ; so that in equal times energy is com-
municated in direct proportion to that speed. When
there is no motion, no energy is communicated ; and this
would certainly not be the case if communication of
energy corresponded to the time during which the tension
was said to act.

12. The muscular sense is far more deceptive than any
other, except, perhaps, that of touch. Conjurors, ven-
triloquists, perfumers, and cooks make their livelihood by
practising on the imperfections of our senses of sight,
hearing, smell, and taste respectively. But he who has
tried the simple experiment of rolling a pea on the table
lietween his first and second fingers, after crossing one-
over the other, will at once recognise the extreme deceit-
fulness of the sense of touch. And the muscular sense

INTRODUCTORY. 11

well deserves a place beside it. So, as we know that
there is but one pea, though the sense of touch vividly
impresses us with the notion that there are two, we must
be very wary when the muscular sense plainly suggests to
us the notion of force as an objective reality.

13. Many of the terms which are now used in a strictly
scientific sense had a humbler origin, having been devised
entirely for the popular expression of common ideas. The
term Work is a specially illustrative one. Thus, in a
draw-well, the work done in bringing water to the surface
would be reckoned at first in terms of the quantity of
water raised: two raisings of a full bucket lifting twice
as much water as one. But then it was found that, for
the same quantity of water raised, the work depended on
the depth of the well : doubled depth corresponding to
doubled work. Again, if the bucket were filled with sand
instead of water, more work was required, in proportion
as sand is heavier than water. All these statements were
soon found to be comprehended in the simple form :
the work done is directly proportional to the weight
raised and also to the height through which it is raised.
Here the indications of the muscular sense stepped in, and
work came to have a general meaning, viz. the product
of the so-called force exerted, into the distance through
which it is exerted.

Had they not possessed the muscular sense, men might
perhaps have been longer than they have been in recog-
nising the important thing potential energy ( 4) ; but
when they had come to recognise it, they would have
stated that when water is raised it gains potential energy
in proportion as it is raised, and perhaps they might have
found it convenient to use a single term for the rate at
which such energy is gained per foot of ascent. This

12 PROPERTIES OF MATTER.

would probably not have been the word " Force," but it
would have expressed precisely what the word force now
expresses.

Then they would have recognised that when energy is
transmitted by a driving-belt, the amount transmitted is
(ceteris paribus) directly proportional to the space through
which the belt has run. They might have invented a
name for the rate of transmission per foot-run of the belt ;
they might even have called it the tension of the belt ; but,
anyhow, it would be precisely what is now called force.

Let us look at the matter from another point of view.

14. A stone, if let fall, gradually gains kinetic energy
or energy of motion, and experiment shows that the
energy gained is directly proportional to the vertical space
fallen through. Hence we have come to say that the
stone is acted upon by a force (its weight, as we call it)
whose amount is practically the same at all moderate
distances from the earth's surface.

But, so far as we know the question scientifically, we
can say no more than that the stone has potential energy
(just as water in a mill-pond has head) in proportion to
its elevation above the earth's surface ; and consequently,
by the conservation of energy, it must acquire energy of
motion in proportion to the space through which it
descends. Why it has potential energy when it is raised,
and why that potential energy takes the first opportunity
of transforming itself into kinetic energy : thus requiring
that the stone shall fall unless it be supported : are
questions to be approached later. (Chap. VII.)

15. That the statement above is complete, without the
introduction of the notion of force, is seen from the fact
that a knowledge of the kinetic energy acquired, after a
given amount of descent, enables us to determine fully

INTRODUCTORY. 13

the nature of the resulting motion even when the stone is
projected, obliquely or vertically, not merely allowed to
fall. The question is easily 'reduced to one of mathe-
matics, or rather of Kinematics, and as such the non-
mathematical student must, for the present, simply accept
the statement as true.

And thus we have another of the many distinct and
independent proofs that Force is a mere phantom sugges- ,
tion of our muscular sense ; though there can be no doubt
that, in the present stage of development of science, the
use of the term enables us greatly to condense our
descriptions.

But it is a matter for serious consideration whether we
do not connive at a species of mystification by thus
employing, in the treatment of objective phenomena, a
term for a mere sensation, corresponding to nothing
objective: even although it be employed solely to shorten
our statements or our demonstrations.

Every one knows that matter (e.g. corn, gold, diamonds)
has its price ; so (as we saw in 7) has energy. We are
not aware of any case in which force has been offered for
sale. To " have its price" is not conclusive of objectivity,
for we know that Titles, Family Secrets, and even Degrees,
are occasionally sold ; but " not to have its price " is at
least all but conclusive against objectivity.

We are in the habit of speaking of fresh air, sunshine,
&c., though obviously objective, as priceless !

16. These introductory remarks have been brought
in with the view of warning the reader that we are
dealing with a subject so imperfectly known that at almost
any part of it we may pass, by a single step as it were,
from what is acquired certainty to what is still subject
for mere conjecture

14 PROPERTIES OF MATTER.

An exact and adequate conception of matter itself,
could we obtain it, would almost certainly be something
extremely unlike any conception of it which our senses
and our reason will ever enable us to form. Our object,
therefore, in what follows, is mainly to state experimental
facts, and to draw from them such conclusions as seem to
be least unwarrantable.

17. But, for the classification of the properties of
matter, whether our classification be a good one or not,
it is necessary that we should have a definition of matter.

From what was said in last section it is obvious that
no definition we can give is likely to be adequate. All
that we can attempt, then, is to select a definition which
(while not obviously erroneous) shall serve as at least a
temporary basis for the classification we adopt.

18. Numberless definitions of matter have been pro-
posed. 1 Here are a few of the more important :

(a) That which possesses Inertia ( 9).

((3) The Receptacle or Vehicle of Energy ( 8).

(y) Whatever exerts or can be acted on by Force.

(8) Whatever can be perceived by our senses, especi-
ally the sense of Touch. This is closely akin
to the well-known definition of matter as a
Permanent Possibility of Sensation.

() Whatever can occupy space.

() Whatever, in virtue of its motion, possesses Energy.

(rj) Whatever, to set it in motion, requires the ex-
penditure of Work.

(0) [Torricelli, Lezioni Accademiche, 1715, p. 25.] La
materia altro non e, che un vaso di Circe incan-
tato, il quale serve per ricettacolo della forza, e de'

1 A remarkable collection of such (now historical) speculations,
due to Professor Flint, is given in Appendix I.

INTRODUCTORY. 15

momenti dell' impeto. La forza poi, e gl' impeti,
sono astratti tanto sottili, son quintessenze tanto
spiritose, che in altre ampolle non si posson
racchiudere, fuor che nell' intima corpulenza de'
solidi natural!.

(0 [The Vortex Hypothesis of Lord Kelvin.] The
rotating parts of an inert perfect fluid ; whose
motion is absolutely continuous, which fills all
space, but which is, when not rotating, absolutely
unperceived by our senses.

|(/c) Surfaces of misfit in a granular medium. [Hypo-
thesis of Osborne Reynolds.]
(A) Loci of intrinsic rotational strain in the ether.

[Hypothesis of Larmor.] See App. V.
(/A) Fundamentally of electric constitution. [J. J.
Thomson and Kelvin.] See Chap. XV. and
App. VI. , ; It should be noted that, as Larmor
has pointed out, the vortex hypothesis gives no
account of electric action. W. P.]
19. The mutual incompatibility of certain pairs of
these definitions shows that some of them, at least, must
be of the so-called metaphysical species ( 3).

(a), (/?), (), (rj), above, have much in common, and,
with further knowledge, may perhaps be found to differ
in expression merely. At present, from want of informa-
tion, we cannot be certain that any two of them are
precisely equivalent.

Berkeley virtually asserted that all motion is produced
by the direct action of spirits on matter. Even then, the
statement (/3) that matter is the receptacle or vehicle of
energy holds good (but how then does energy exist in
the spirit ?).

But the statement that matter is whatever can exert

16 PROPERTIES OF MATTER.

force (y) is to be rejected ; though it was virtually intro-
duced by Cotes in his Preface to the second edition of
the Principia.

(8) must be rejected, if only because there is another
thing besides matter (in the physical universe) which we
know of, and of course only through our senses ( 1).
But this is not all the error ; for we get the notion of
force through our muscular sense ( 11), and force is not
matter, not even a thing.

Torricelli's language is poetical, and therefore his
statement (6) must not be taken too literally. In his
time, as in all subsequent time till well within the last
half century, energy and force were very rarely distin-
guished from one another. Even now they are too often
confounded.

(i), the most recent of these speculations, has the
curious peculiarity of making matter, as we can perceive
it, depend upon the existence of a particular kind of
motion of a medium which, under many of the defini-
tions above, would be entitled to claim the name of
matter, even when it is not set in rotation.

20. But as we do not know, and are probably incapable
of discovering, what matter is, we must content ourselves
for the present with a definition which, while not at
least obviously incorrect, shall for the time serve as a
working hypothesis.

"We therefore choose (e) above, i.e. we define, for the
moment, as follows :

Matter is whatever can occupy space.

Experience has proved that it is from this side that the
average student can most easily approach the subject, i.e.
here, as it were, the contour lines of the ascent ( 80) are
most widely separated.

INTRODUCTORY. 17

21. A But this definition involves three distinct proper-
ties: (1) the Volume, (2) the Form or Figure, of the space
occupied ; and (3) the nature or quality of the Occupation.

Hence the older classical works on our subject almost
invariably speak of matter as possessing (1) Extension,
(2) Form, and (3) Impenetrability. It is mainly for the
sake of the first of these, and the preliminary discussions
which it necessarily introduces, that we have chosen the
above definition as our starting-point.

22. Before we take these up in detail, however, it may
be useful to devote a short chapter to a digression on
some of the more notable of the hypotheses which have
been propounded as to the ultimate structure of matter.
We advisedly use the word structure instead of nature,
for it must be repeated, till it is fully accepted, that the
discovery of the ultimate nature of matter is probably
beyond the range of human intelligence.

Another chapter, of a very miscellaneous character, will
follow, devoted to the examination of some of the terms
popularly applied to pieces of matter, and a rapid glance
at the physical truths which underlie them. This is
introduced to give the reader, at the very outset of his
work, a general idea of its nature and extent.

CHAPTER II.

SOME HYPOTHESES AS TO THE ULTIMATE STRUCTURE OF
MATTER.

23. THE hard Atom, glorified in the grand poem of
Lucretius, but originally conceived of, some 2400 years ago,
by the Greek philosophers Demokritus and Leukippus,
survives (as at least an unrefuted, though a very improb-
able, hypothesis) to this day. Newton made use of the
hypothesis of finite, hard, atoms to explain why the speed
of sound in air was found to be considerably greater than
that given by his calculations ; which were accurate in
themselves, but founded on erroneous or, rather, incom-
plete data. But in this problem Laplace found the vera
causa, and in consequence Newton's apparent support of
the hypothesis of hard atoms is no longer available.

Many of the postulates of this theory are with difficulty
reconciled with our present knowledge ; some have been
contemptuously dismissed as "inconceivable." But any
one who argues on these lines becomes, ipso facto, one of
the so-called metaphysicians.

Let us briefly consider the main statements of this
theory, but without regard to the order in which
Lucretius gives them.

24. Nature works by invisible things : thus paving-

ULTIMATE STKUCTURE OF MATTEK. 19

stones and ploughshares are gradually worn down without
the loss of any visible particles.

Reproduction [i.e. agglomeration of scattered particles
so as to produce visible bodies] is slower than decay
[i.e. the breaking up of bodies into invisible particles], and
therefore there must be a limit to breakage, else the
breaking of infinite past ages would have prevented any
reproduction within finite time. Hence there exists a
least in things [i.e. unbreakable parts or Atoms, "strong
in solid singleness "].

But there is also void in things, else they would be
jammed together, and unable to move. Here Lucretius
takes the case of a fish moving in water, showing that
void is necessary in order that it may be able to move.
[Our modern knowledge of circulation, i.e. the motion of
fluids in re-entrant paths, shows that this reasoning is
baseless.]

There can be no third thing besides body and void.
For nothing but body can touch and be touched; and
what cannot be touched is void. [Here we have the germ
of the erroneous definition of matter (8) in 18 above.]

The atoms are infinite in number, and the void in
which they move [space] is unlimited.

They have different shapes ; but the number of shapes
is finite, and there is an infinite number of atoms of each
shape.

Nothing whose nature is apparent to sense consists of
one kind of atoms only.

The atoms move through void at a greater speed than
does sunlight.

Besides this, there is a great deal of curious speculation
as to bow a vertical downpour of atoms [supposed to be
a result of their weight] is, in some arbitrary way, made

20 PROPERTIES OF MATTER.

consistent with their meeting one another and agglome-
rating into visihle masses of matter.

The basis of the whole of Lucretius' reasoning in
favour of the existence of atoms lies in the gratuitous
assumption that reproduction is slower than decay. This
is by no means consistent with our modern knowledge,
for potential energy of different masses [whether gravita-
tional or chemical] is constantly tending to the agglomera-
tion of parts, and on a far grander scale than that in which
any known cause tends to decay or breaking up.

But if there be hard atoms, they must (in all known
bodies) have intervals between them ; for compressi-
bility : i.e. capability of having the component atoms
brought more closely together : is a characteristic of all
known bodies. [Contrast this mode of arriving at the
conclusion that " there must be void in things," with the
erroneous mode employed by Lucretius.]

25. A refinement of this theory, mainly due to
Boscovich, gets rid of the material atom altogether,
substituting for it a mere mathematical point, towards
or from which certain forces tend. It is supported by
the assertion that we know matter only by the effects
which it produces (or seems to produce), and therefore
that, if these effects can be otherwise explained, we need
not assume the existence of substance or body at all.
This theory was, at least in part, accepted by so great an
experimenter and reasoner as Faraday. It virtually
substitutes force for matter as an objective thing ( 2) }
and it essentially involves the heresy of distance-action
( 10). But the fatal objection to which it is exposed is
that it does not seem capable of explaining inertia, which
is certainly a distinctive (perhaps the most distinctive)
property of matter

ULTIMATE STRUCTURE OF MATTER. 21

This theory must be regarded as a mere mathematical
fiction, very similar to that which (in the hands of
Poisson and Gauss) contributed so much to the theory
of statical Electricity ; though, of course, it could in no
way aid inquiry as to what electricity is.

26. A much more plausible theory is that matter is
continuous (i.e. not made up of particles situated at a
distance from one another) and compressible, but in-
tensely heterogeneous ; like a plum-pudding, for instance,
or a mass of brick-work. The finite heterogeneousness
of the most homogeneous bodies, such as water, mercury,
or lead, is proved by many quite independent trains of
argument based on experimental facts. If such a con-
stitution of matter be assumed, it has been shown l that
gravitation alone would suffice to explain at least the
greater part of the phenomena which (for want of know-
ledge) we at present ascribe to the so-called Molecular
Forces. But it does not seem to be compatible with
experimental facts; especially some of the simpler
phenomena presented by gases. ( 55, 322.)

27. The most recent attempt at a theory of the
structure of matter, the hypothesis of Vortex Atoms, is
of a perfectly unique, self-contained character. Its postu-
lates are few and simple, but the working out of anything
beyond their immediate consequences is a task to tax to
the utmost the powers of the greatest mathematicians for
generations to come. A vortex filament, in a pefect
fluid, is a true " atom ; " but it is not hard like those of
Lucretius ; it cannot be cut, but that is because it
necessarily wriggles away from the knife.

The idea that motion is, in some sort, the basis of
what we call matter is an old one : but no distinct con-
1 "W. Thomson, Pro c . R.S.E. t 1862.

22 PROPERTIES OF MATTER.

ceptions on the subject were possible until v. Helmholtz,
in 1858, made a grand contribution to hydrokinetics in
the shape of his theory of vortex motion. 1 He proved,
among other entirely novel propositions, that the rotating
portions of a continuous incompressible fluid, in which
there is neither viscosity nor finite slipping, maintain
their identity : being thus for ever definitely differenti-
ated from the non-rotating parts. He also showed that
these rotating portions are necessarily arranged in con-
tinuous, endless filaments : forming closed curves, which
may be knotted or linked in any way : unless they
extend to the bounding surface of the fluid, in which
alone they can have ends. Thus, to give ends to a
closed vortex filament (i.e. to cut it) we must separate
the fluid mass itself, of which it is a portion : so that
on Lord Kelvin's theory we must (virtually) sever space
itself.

Such vortex filaments (though necessarily of an im-
perfect character) are produced when air is forced to
escape from a box, through a circular hole in one side,
by sharply pushing in the opposite side. If the air in
the box be filled with smoke, or with sal-ammoniac
crystals, the escaping vortex ring is visible to the eye;
and the collisions of two vortex rings, which rebound
from one another, and vibrate in consequence of the
shock, as if they had been made of solid india-rubber,
are easily exhibited. Experimental results of this kind
led Lord Kelvin 2 to propound the theory that matter,
such as we perceive it, is merely the rotating parts of a
fluid which fills all space. This fluid, whatever it be,
must have inertia : that is one of the indispensable

1 Crelle, 1858. Translated (by Tait) in Phil. Mag., 1867.

2 Proc. R.S.E., 1867.

ULTIMATE STKUCTURE OF MATTER. 23

postulates of v. Helmholtz's investigation ; and the great
primary objection to Lord Kelvin's theory is, that it
explains matter only by the help of something else which,
though it is not what we call matter, must possess what
we consider to be one of the most distinctive properties
of matter.

28. This theory is still in its infancy, and we cannot as
yet tell whether it will pass with credit the severe ordeal
which lies before it, when the properties of vortices (which
must be discovered by mathematical investigation) shall
be compared, one by one, with the experimentally ascer-
tained properties of matter. As we have already said,
this theory is self-contained ; no new hypotheses can be
introduced into it ; so that it possesses, as it were, no
adaptability, or capability of being modified, but must fall
before the very first demonstrated insufficiency, or contra-
diction, if such should ever be discovered.

29. But the really extraordinary fact, already known in
this part of our subject, is the apparently perfect similarity
and equality of any two particles of the same kind of gas,
probably of each individual species of matter when it is
reduced to the state of vapour. Of such parts, therefore,
whether they be further divisible or not, each species of
solid or liquid must be looked on as built up. This
similarity of parts, very small indeed but still of essenti-
ally finite magnitude, has been so well treated by Clerk-
Maxwell that, instead of insisting upon it here, we give a
considerable extract from one of his remarkable articles
in Appendix II. below.

30. The further treatment of the subject of structure,
involving the question of how the component parts (be
they atoms or not) of bodies are put together, must be
deferred to the end of the work. What has been said

24 PROPERTIES OF MATTER.

above must be looked on as a mere preliminary sketch,
not intended even to be fully understood until the
experimental data, on which all our reasoning must be
based, are brought before the reader as completely as our
limits permit.

CHAPTER III.

EXAMPLES OF TERMS IN COMMON USE AS APPLIED TO
MATTER.

31. BEFORE we proceed to a more rigorous treatment of
our subject, it may be well to consider what physical truth
underlies each of some of the many adjectives in common
use as applied to portions of matter, such as Massive,
Heavy, Plastic, Ductile, Viscous, Elastic, Rigid, Opaque,
Blue, Coherent, etc.

This course secures a twofold gain, so far as the
beginner is concerned ; for, first, he is introduced by it, in
a familiar way, to some of the more important terms
which are indispensable in scientific description ; and
second, he obtains a glance here and there through the
whole subject of Natural Philosophy, because the pro
gramme before us is so vague as to leave room for
innumerable digressions, each introducing some novel but
important fact or property. But we must endeavour to
be brief, for whole volumes would have to be written
before this subject could be nearly exhausted.

32. Every one who has used his senses to any purpose
knows, before he comes to the study of our science, a
great many of its phenomena, among them some of the
yet unexplained. But he knows, as it were, each by

26 PROPERTIES OF MATTER.

itself, and only in its more prominent features; the
analysis of the appearances or impressions which he has
seen and experienced, and the explanation of the physi-
cal fact or process which underlies each of them, are
absolutely necessary before he can understand the mode
in which they must be grouped, and the reasons for such
grouping.

33. Thus he knows that the moon keeps company with
the earth, never receding nor approaching by more than a
small fraction of the average distance. He also knows
that the earth keeps, within narrow limits, at a definite
distance from the sun. He has a general notion, at least,
that the state of matters on the earth would become
serious, as regards both animal and vegetable life, if we
were to approach to even half our present distance from
the sun, or recede to double that distance. But he would
require to be a Newton if, without instruction, he could
divine that these results are due to the very cause which
keeps the bob of a conical pendulum moving in a horizontal
circle.

He sees ripples running along on the surface of a pool,
but requires to be told that their motion depends upon
the cause which rounds the drops of water on a cabbage-
blade, or in a shower, and which renders it almost
impossible to keep a water-surface clean.

He sees what he calls a flash of lightning, but he
requires to be told that what he sees is mainly particles
of air heated so as to be self-luminous.

He looks at the stars and thinks he sees them as they
are, but he requires to be informed that he fees even the
nearest of them only as it was three years ago and that it
may have changed entirely in the interval.

And he will certainly require to be informed, even

TERMS APPLIED TO MATTER. 27

with patient iteration, that air is made up of separate and
independent particles : the number of which in a single
cubic inch is expressed by twenty-one places of figures, a
multitude altogether beyond human conception : a busy
jostling crowd, each member of which darts about in
all directions, impinging on its neighbours some eight
thousand million times per second.

But when he has got so far, and has been told that
this astounding information is as nothing to what we
feel convinced that science can yet reveal, he cannot help
marvelling alike at the arcana of physics, and at the
patient efforts of genius which have already penetrated
so far into the darkness shrouding its mysteries.

34. Take the terms Massive and Heavy as applied to
a piece of matter, or the corresponding substantives, the
Mass and the Weight of a body.

The terms are usually regarded as equivalents, but in
their origin they are completely distinct. The one is a
property of the body itself, and is retained by it without
increase or diminution wherever in the universe the body
may be situated. The other depends for its very exist-
ence on the presence of a second body, the Earth : and
varies in inverse proportion to the square of the distance
from the earth's centre.

The destructive effects of a cannon-ball are due entirely
to its mass and to the relative speed with which it
impinges on the target. They have nothing to do with
its weight, for they would be exactly the same (for the
same relative speed) in regions so far from the earth, or
other attracting body, that the ball had practically no
weight at all.

When an engine starts a train on a level railway, or
when a man projects a curling-stone along smooth ice,

28 PROPERTIES OF MATTER.

the resistance which either prime mover has to overcome
is due to the mass of the body to be moved. Its weight,
except indirectly through friction, has nothing to do with
it. So when we open a large iron gate properly sup-
ported on hinges, it is the mass with which we have to
deal ; if it were lying on the ground and we tried to lift
it, we should have to deal simultaneously with its weight
and with its mass.

The exact proportionality of the weights of bodies to
their masses, at any one place on the earth's surface,
was proved experimentally by Newton, and is thus no
mere truism, but an essential part of the great law of
gravitation.

Thus a pound of matter is a definite amount, or mass,
of matter, unchangeable whithersoever that matter may
be carried. But the weight of a pound of matter, or a
" pound- weight," as it is commonly called, is a variable
quantity, depending upon the position of the body with
respect to the earth ; and changes (to an easily measurable
amount) as we carry the body to different latitudes, even
without leaving the sea-level.

35. The common use of the balance as a means of
measuring out equal quantities of matter is justified by
Newton's result ; but the process is essentially an indirect
one, for the balance tells only of equality of weight. If
the earth were hollow at the core, the balance would
cease to act in the cavity ( 140). Bodies would preserve
their masses there, but would be deprived of weight, at
least so far as the earth is concerned.

To sum up for the present, the mass of a body is
estimated by its inertia, and is taken as the measure of
the amount of matter in the body ; while the weight is
an accidental property, connected with the presence of

TEEMS APPLIED TO MATTER. 29

another mass of matter. But it is a most remarkable
fact that under the same given external conditions the
weight depends upon the quantity only, and not on the
quality, of the matter in a body.

If a body, A, becomes heavy in consequence of the
presence of another body, B, so in like wise does B become
heavy in consequence of A's presence. And the weights
of the two, each as produced by the attraction of the
other, are exactly equal. Hence, if they be free to
move, the quantities of motion (i.e. the momenta) pro-
duced in a given time are equal and opposite. [Newton's
Lex iii. 128.] But as the momentum is the product of
the mass and the velocity, the parts of the velocities of
the two bodies, due to their mutual gravitation alone,
will be in amount inversely as their masses. Thus,
though the weight of the whole earth, produced by the
attraction of a stone, is exactly equal to that of the stone
produced by the attraction of the earth, the consequent
rate of fall of the earth towards the stone is less than
that of the stone towards the earth in the same ratio that
the mass of the stone is less than that of the earth, and
is therefore usually so small as to escape observation.
The moon, however, is a stone whose mass is not exces-
sively smaller than that of the earth, and the consequences
of the earth's fall towards the moon have to be taken
account of in astronomy.

36. To properties such as mass, which depends on the
size as well as on the material of a body, and weight
which, in addition, depends on a second body, there
correspond what are called specific properties, characteristic
of the substance and independent of the dimensions of the
particular specimen examined.

Thus the mass of a cubic foot of any kind of matter

30 PROPERTIES OF MATTER.

may be called its specific mass. But this quantity, i.e.
the amount of matter in unit bulk, is usually expressed
by the term Density.

The weight of a cubic foot of each particular kind of
matter in any locality may be called the specific weight.
But as this varies, though in the same proportion for all
bodies, from place to place, we use instead of it the ratio
of the weight of a cubic foot of the substance to that of
a cubic foot of some standard substance. This is called
the Specific Gravity. Pure water, at the temperature
called 4 C. (its maximum-density point), is usually taken
as the standard substance.

Newton's experimental result shows that the density
and the specific gravity of any substance are proportional
to one another, so that if the density of water at 4 C.
l)e taken as unit-density, a table of specific gravities is
identical with a table of densities. ( 166.) But we
must repeat, the coincidence is an experimental fact, not
(as yet at least) in any sense a truism.

Specific gravity is, in general, much more easily
measured with accuracy than is density, so that it is usually
the property to be directly determined, the other simply
following from it in consequence of Newton's discovery.

37. To vary the subject widely, let us now consider
the term Viscous as applied to fluids. The contrasted
adjective is usually taken as Mobile.

When a liquid partially fills a vessel, and has come to
rest, it assumes a horizontal upper surface. If the vessel
be tilted, and held for a time in its new position, the
liquid will again ultimately settle into a definite position,
with its surface again horizontal. Practically it occupies
the same bulk in each of these positions. Hence the only
change it has suffered is a chsnge of /

TERMS APPLIED TO MATTER.

31

But this change of form is much more rapidly attained
in some liquids than in others, even v/hen they are of
nearly the same density. Some (such as sulphuric ether)
attain their equilibrium position so quickly that they
retain energy enough to oscillate about it for some time
before coming to rest; others (such as treacle) attain it
only after a long time and, unless in great masses and
when violently disturbed, do not oscillate but gradually
creep to their final shape. Hence we call treacle viscous.
To analyse this result let us consider (in a very ele-
mentary case, for the general analysis of the process
requires higher mathematical methods than we can employ
in a work like this) what is involved in Shear: i.e.
change of form of a body without change of bulk.

38. When water flows, without eddies, slowly in a
rectangular channel of uniform width and depth, we
know, by observation of particles suspended in it, that
the upper parts flow faster than the lower, and (practically)
in such a way that a column of the water, originally
straight and vertical, inclines, as a whole, forwards more
and more in the direction of its motion. Hence in a
vertical section, along the middle of the channel, the
particles originally forming the line ab in the figure will,

c a' c'

6 d V d,'

. FIG.1.,

after the lapse of a certain time, be found approximately
in the line a'b'. Similarly those which were originally in

32 PROPERTIES OF MATTER.

cd parallel to ab, will be found in c'd', parallel to a'b', and
so situated that a'c' = ac, and of course also b'd' = bd. The
figures ad, a'd", are thus parallelograms on equal bases
and between the same parallels, and therefore equal in
area. This shows that the water enclosed between
vertical cross sections through ab and cd has the same
volume as that between inclined sections (perpendicular
to the sides) passing through a'b' and c'd'. There has
thus been change of form only in this mass of water, and
we see that it has been produced by the sliding of every
horizontal layer of the water over that immediately
beneath it. [The same result follows even if a'b' be not
straight, for c'd' will necessarily be equal and similar to
it.] A good illustration of the nature of this kind of
distortion will be seen in the leaves of an opened book,
especially a thick one, such as the London Directory. It
is often well exhibited by piles of copies of a pamphlet,
or of quires of note-paper curiously arranged in a shop-
window. Now when there is resistance to sliding of one
solid on another we call it Friction. Thus the viscosity
of a fluid is due to its internal friction, just as the slower
motion at the bottom than at the top of the channel is
to be ascribed to the friction of the liquid against the
solid. [Another illustration of the subject is frequently
furnished by the way in which we can judge of the
direction of the wind from the mere form of detached
clouds, as seen at a single glance. For such clouds, if
originally nearly spherical, are distorted into ellipsoids
whose longer axes overhang, as it were, the direction of
motion.]

39. We now see why it is that disturbances of liquids
gradually die away : why the waves on a lake, or even
on an ocean, last so short a time after the storm which

TERMS APPLIED TO MATTER. 33

produced them has ceased. Also why winds (for there
is friction in gaseous fluids as well as in liquids, though
the mechanical explanation of its origin may not be quite
the same) gradually die out. In either case the energy
apparently lost is, as in the case of friction of solids,
merely transformed into heat. We also see why it is
that winds have the power of raising water-waves.

The stirring of water, or oil, and the measurement of
the consequent rise of temperature when the whole had
come to rest, the work done in stirring being also deter-
mined, was one of the processes by which Joule found,
with great accuracy, the dynamical equivalent of heat.

40. It is very instructive to watch the ascent of an
air-bubble in glycerine, and to compare it with that of
an equal bubble in water. The experiment is easily
tried with long cylindrical bottles, nearly full of different
liquids, but having a small quantity of air under the
stopper. When the bottle is inverted the bubble has to
traverse the whole column.

The (apparent) suspension in water of mud, and ex-
ceedingly fine sand (to whose presence the exquisite
colours of the sea and of Alpine lakes are mainly due), is
merely another example of viscosity. So is the suspen-
sion of fine dust, and of cloud particles, in the air.
Stokes 1 calculates that a droplet of water, a thousandth
of an inch in diameter, cannot fall in still air at a much
greater rate than an inch and a half per second. If it be
of one-tenth of that size it will fall a hundred times
slower, i.e. not more than one inch per minute ! It is
very remarkable that the resistance in such cases varies
as the diameter of the drop. (See 316.) With larger

1 On the Effect of the internal Friction of Fluids on the Motion
of Pendulums. Camb. Phil. Trans, ix. (1851), eq n (127).

34 PROPERTIES OF MATTER

bodies, moving faster, the resistance is proportional to the
sectional area, being then in great part due to another
cause than viscosity.

41. Bodies are called Elastic or Non-elastic. Compare,
for instance, the properties of a wire of steel with those
of a lead wire ; or of a piece of india-rubber and a piece
of clay or putty. But the popular use of these terms is
generally very inaccurate. The blame rests mainly with
the ordinary text-books of science, which are (as a rule)
singularly at fault with regard to the whole of this special
subject, including even its most elementary parts.

Elasticity, in the correct use of the term, implies that
property of a body in virtue of which it recovers, or tends
to recover, from a deformation.

The phrase " tends to recover " is scarcely scientific ;
we should preferably say " requires the continued applica-
tion of deforming stress ( 128) to prevent recovery, entire
or partial, from deformation."

Kinematics shows us that any deformation, however
complex, is made up of mere changes of bulk and of form.

A distortion may therefore be wholly Compression, or
wholly Shear ( 37), or made up of these in any way.
Hence there are two distinct kinds of Elasticity, viz.
Elasticity of Bulk and Elasticity of Form. The former
is possessed in perfection by all fluids, while the second
is wholly absent. In solids both are present, but neither
in perfection (except perhaps in very special cases, and
then within very narrow limits).

Thus we see that, as a necessary preliminary to in-
vestigations on elasticity of bodies, we must study their
capabilities of being distorted : a whole series of pro-
perties, such as compressibility, extensibility, rigidity, etc.

This investigation is given in Chap. VIII., and its
applications in Chaps. IX., X., XL below.

TERMS APPLIED TO MATTER. 35

42. In popular language, bodies are said to be White,
Black, Blue, Red, etc. The investigation of the under-
lying scientific facts, on which these depend, is partly
physical (and therefore within our scope), but also partly
physiological. The subject is thus a somewhat complex
one.

What do we mean by White Light ? This is a question
much more physiological than physical ; dealing, as it
does, with phenomena which are subjective rather than
objective. Probably the true answer to it depends upon
circumstances, or conditions, which may be varied in-
definitely, and with them will necessarily vary what is
described in terms of them.

Thus, in a room lit by gas, a piece of ordinary writing-
paper, or of chalk, appears white : at least if we have
been in the room for some little time. But if, beside it,
there be another piece of the same paper or chalk on
which, through a chink, a ray of sunlight is allowed to
fall (weakened, if necessary, so as to make the two appear
of nearly the same brightness), we at once call the first
piece of paper or chalk yellow, allowing the second to be
white. Here we enter on a purely physiological question.
In fact, if we accustom ourselves, for a sufficiently long
time, to the observation of bodies in a room lit up only
by burning sodium (which gives almost homogeneous
orange light), we may ultimately come to regard bright
bodies such as chalk, etc., as being white : others, of
course, being merely of different shades, or degrees of
blackness. This, therefore, is foreign to our present
subject. For all that, it furnishes us with the means of
answering an important question somewhat different from
that proposed above, but now a physical question : viz.
What do we mean by a white body ?

36 PROPERTIES OF MATTER.

43. Suppose two sources to exist in the room, giving
different kinds of homogeneous light; one being incan-
descent sodium as before, the other incandescent lithium,
which (at moderate temperatures) gives a homogeneous red
light. Chalk and ordinary writing-paper will still appear
as white bodies to an eye which has become accustomed
to the light in the room ; other bodies appear darker, but
some are reddish, some of an orange tint.

And thus we obtain the idea that what we call a white
body is one which sends to the eye, in nearly the same
proportion as it receives them, the various constituents
of the light which falls upon it ; while a black body
sends none ; and coloured bodies send back light which,
while (in general) necessarily made up of the same con-
stituents as the incident light, contains them in different
proportions to those in which they fell upon it. [It
would only confuse the student were we here to refer to
Fluorescence.]

44. Thus white light would seem to be a mere relative
term. It is conceivable that the inhabitants of worlds
whose sun is a blue star, or a red star (and there are
many notable examples of such stars), may have their
peculiar ideas of white light, formed from their own
circumstances ; as ours is formed from the light of our
own sun, which is what, in contrast with these, we must
call a yellow star.

However this may be, the discussion above has shown
what is meant by a white body. A blue body is, by
similar reasoning, one which returns blue rays in greater
proportion than it does those of other visible light. It is
therefore said to absorb the other rays in greater proportion
than it absorbs the blue rays.

Now we are in a position to understand why blue and

TERMS APPLIED TO MATTEfc. 37

yellow pigments, mixed together, give green : while a
disc, painted with alternate sectors of the same blue and
yellow, appears of a purplish colour when made to rotate
rapidly. .For the light given out by the rotating disc is
a mixture, in the proportion of the angles of the sectors, of
the kinds of light returned by the blue and yellow separ-
ately. But that which the mixed pigments send back
has in great part penetrated far enough into the mass to
run, as it were, the gauntlet of absorption by each of the
separate components in turn, and therefore is finally made
up of those rays alone which are not freely absorbed by
either.

To this discussion we need only add, in illustration of
the conservation of energy, that a body is always found
to be heated in proportion to the amount of light-energy
which it absorbs.

45. Shifting our ground again, we next take the words
Malleable, Ductile, Plastic, and Friable, as applied to
solid bodies.

All of these refer specially to the behaviour of solids
under the action of stresses which tend to change their
form; for the change of volume of solids, even under
very great pressures, is usually very small. The first
three indicate that the body preserves its continuity while
yielding to such stresses, the fourth that it breaks into
smaller parts rather than allow its form to be changed.
And, in popular use at least, the terms imply in addition
that the body is not sensibly elastic.

46. The most perfect example of a malleable body is
metallic gold. The gold leaf employed for " gilding," as
it is called, is prepared by a somewhat tedious process,
which requires a high degree of skill in the workman.
The gold is first rolled into sheets thinner than the thin-

38 PROPERTIES OF MATTER.

nest writing-paper (thus already showing a high amount of
plasticity) ; next it is beaten out between leaves of vellum,
till its surface is increased, and therefore its thickness
diminished, some twenty-fold. A small portion of this
fine leaf is then placed between two pieces of gold-
beater's skin ; and a more skilful workman, with a lighter
hammer, again extends its surface twenty-fold. This
operation can be repeated without tearing the thin film of
metal, so great is its tenacity. ( 226.)

Here we have one dimension (thickness) diminished
in a marked manner, but the product of the other two
dimensions (the surface of the leaf) is of course pro-
portionally increased.

47. The action of the hammer may be practically
viewed as equivalent to that of an intense pressure exerted
through a very small volume, thus at every stroke apply-
ing a finite amount of energy. One portion of this is
changed into heat in the hammer, the anvil, and the
gold leaf ; the rest is employed in doing work against the
molecular forces of the gold, and thus altering its form.

To show that this is the true explanation of the
observed effect, we may vary the experiment by subject-
ing a leaden bullet to the action of a hydrostatic press.
A few strokes of the pump suffice to bruise the bullet
into a mere cake. The process is essentially the same as
that of gold-beating, but lead is by no means so malleable
as gold.

48. This leads us, in our present discursive treatment
of parts of our subject, to inquire how it is, that by
means of such a machine as the Bramah press, a man can
apply pressure sufficient to mould a piece of lead, whose
shape he could scarcely alter to a perceptible amount by
the direct pressure of the hand.

TERMS APPLIED TO MATTER. 89

Here we have a first inkling of the Function of a
Machine. A machine is merely a contrivance by which
we can apply work in the way most suitable for the
purpose we have in hand. Work (as a form of energy)
is a real thing, whose amount is conserved. But we
have seen that it can be measured as the product of
two factors the (so-called) force exerted, and the space
through which it is exerted. Hence, because even when
a machine is perfect it can give out only the energy
communicated to it, if there be but one movable part
to which energy is supplied and another by which it is
given off, the simultaneous linear motions of these two
parts must be in the inverse ratio of the forces applied to
them, or exerted by them, in the direction of these
motions respectively. Thus we are not concerned with
the interior structure, or mode of action, of a perfect
machine : all we need to know is the necessary relation
of the speeds of the two parts or places at which energy
is taken in and given out. This is a matter of kinematics,
and can be made the subject of direct measurement when
the machine is caused to move, whether it be transmitting
work or not.

The statement just made is embodied in the vernacular
phrase

What is gained in power is lost in speed.

Objections may freely be taken to this form of words,
but it is meant to imply precisely what was said above
as to the action of a perfect machine.

If the machine be imperfect, as, for instance, if there
be frictional heating during its working, the heat so
produced represents some of the energy given to the
machine, and the remainder of it is alone efficient.

40 PROPERTIES OF MATTER.

49. A substance is said to be ductile when it can be
drawn into very fine wires i.e. when it admits of great
exaggeration of one of its three dimensions (length) at
the expense of the product of the other two (cross
section). Wire -drawing is, essentially, a very coarse
operation, for it has to be effected by finite stages, the
wire being drawn in succession through a number of
holes in a hard steel plate, in which each hole is a little
smaller in diameter than the preceding one. The more
nearly continuous the operation is made, the more tedious
and therefore the more costly it becomes.

The associated tenacity and plasticity of silver render
it one of the most ductile of metals. And an ingenious
idea of Wollaston's enables us, as it were, to impart to
other metals much of the ductility of silver. His idea
may be briefly explained by analogy as follows. Suppose
a glass rod, whose core is coloured, be drawn out while
softened by heating, the diameter of the core is found to
be reduced in the same proportion as is that of the rod.
Thus, to obtain platinum wires much finer than could be
procured by direct drawing, Wollaston suggested the
boring of a hole in the axis of a cylindrical rod of silver,
plugging the hole with a platinum wire which just fitted
it, and then drawing into fine wire the compound
cylinder. When this operation has been carried to its
limit, practically determined by the ductility of the
silver, the diameter of the platinum has been reduced
nearly in the same proportion as that of the silver ; and
the silver may be at once removed from the fine platinum
core by plunging the whole in an acid which freely
attacks silver but has no effect on platinum.

50. Plasticity is shown, on the large scale, by many
substances which, in hand specimens, appear fragile in

TERMS APPLIED TO MATTER. 41

the extreme. Glacier-ice is one of these, but its behaviour
is so closely connected with its thermal properties that
we can only mention it here.

The earth as a whole, though its rock-structure appears
so rigid, has been found to be more plastic (under the
tidal attraction of the moon) than a globe of glass of the
same size would have been.

But it is specially under the action of small but
persistent forces that bodies, which are usually regarded
as brittle or friable, show themselves to be really plastic.
A good example of this is given by an experiment due to
Lord Kelvin. Cobbler's wax is usually regarded as a
very brittle body ; yet if a thick cake of it be laid upon
a few corks, and have a few bullets placed on its upper
surface (the whole being kept in a great mass of water
to prevent any but small changes of temperature), after
a few months the corks will be found to have forced
their way upwards to the top of the cake, while the
bullets will have penetrated to the bottom.

[Tresca has investigated the flow of solid metals, and
has shown that even steel flows under the action of
sufficient stress. W. P.]

51. For variety, let us next take the terms Trans-
parent, Translucent, and Opaque.

These refer, of course, to the behaviour of a substance
with regard to the passage of light through it. In
common speech, a pane of ordinary window-glass is
called transparent, while a piece of corrugated or of
ground glass is translucent : the latter transmits rays,
no doubt, but with their courses so altered that they are
no longer capable of producing distinct vision of the
source from which they come. Consistency would require
that the term translucent should also be applied to

42 PROPERTIES OF MATTER.

irregularly-heated air, or to a mixture of water and strong
brine before diffusion has rendered it uniform throughout.

Translucent is hardly a scientific word, unless we
choose to limit its application to heterogeneous bodies.
In science we speak of the degree of transparency of a
homogeneous substance ; as, for instance, water more or
less coloured, and employed in greater or less thickness.
In such cases, besides the inevitable surface-reflection,
there is more or less absorption ; and the percentage of
any definite kind of incident light which unit thickness
of the substance transmits is called its transparency for
that kind of light.

Opacity may arise from either of the two causes just
mentioned. Light may enter a body, and be unable to
proceed farther, as is the case with lamp black. Or it
may fall on a highly polished surface, such as thinly
silvered glass, and be in great part reflected without
entering.

In the former case it is said to be absorbed ; and, when
this happens, the absorbing body is raised in temperature.
The incident energy is converted from the radiant form
into that of heat.

In the latter case part only can enter the body; and, if
it meet in succession other reflecting surfaces in sufficient
number, practically the whole of it may be reflected.
This is the case with a heap of pounded glass, a cloud,
a mass of snow, or of froth or foam. All of these
materials are transparent, but they reflect some of the
incident light; and, in consequence of the multiplicity
of surfaces which the light has to encounter, the greater
part of it is reflected before it has penetrated deeply into
the mass. Hence the whiteness and brightness of snow
and clouds in full sunshine.

TERMS APPLIED TO MATTER. 43

52. We have here an excellent opportunity of calling
the student's attention to the distinction : a very pro-
found one : between Heat and Temperature.

For we have seen that energy, in the form of light,
when absorbed, becomes heat in the absorbing body, and
thus raises its temperature. But if the same quantity of
heat had been given to a body, of the same nature but
of twice the mass, the rise of temperature would have
been only half as great. The very form of words here
used shows at once how different are the meanings of
the words temperature and heat. For the quantity of
heat (so much energy, a real thing) is perfectly definite,
but the effect it produces on the temperature (a mere
state) depends on the quantity and quality of the mass to
which it is communicated.

Heat is therefore a thing, something objective; tem-
perature is a mere CONDITION of the body with which
the heat is temporarily associated ; a condition which
in certain cases determines the physical state of the body
itself, and in all cases determines its readiness to part
with heat to surrounding bodies or to receive it from
them.

Heat may, in this connection, but only for illustration,
be compared with the air compressed into the receiver of
an air-gun ; temperature would then be analogous to the
pressure of that air. Neither of two receivers would
(except by diffusion, with which we are not at present
concerned) give air to the other, when a pipe is opened
between them, if the pressure were the same in both ;
but air would certainly flow from the receiver in which
the pressure is greater to the other ; and this, altogether
independently of the relative capacities of the two receivers,
or the consequent amounts of their contents.

44 PROPERTIES OF MATTER.

53. As another example, take the terms Cohesive, In-
coherent, Repulsive.

A lump of sandstone has considerable tenacity, which,
of course, is to be ascribed to those molecular forces of
which we spoke in 26. But when, in virtue of its
friability, it has been pounded down into sand, it becomes
an incoherent powder. And we know that it must at
some time previously have been in this form, for it often
contains fossil plants or fish, and it may even have pre-
served (perhaps for a million or more of years) records of
surface-disturbance in the form of dents made by rain
or hail, or by the feet of birds or reptiles. When a
sufficiently deep layer of sand is deposited, by drifting
or otherwise, above this portion, its loose particles are
brought by the consequent pressure so close together that
their molecular forces once more come into play.

The graphite, or plumbago, which forms the material
for the finest drawing-pencils, is a somewhat rare and
valuable mineral. In cutting it up into " leads," however
carefully, a considerable portion is reduced to powder
i.e. sawdust. But if this powder be exposed, in mass, to
pressure sufficient to bring its particles once more within
the extremely short mutual distance at which the molecular
forces are sensible, these forces again come into play, and
the powder becomes a solid mass, which can in turn
be sawn into " leads " for a somewhat inferior class of
pencils.

The whole of this part of the subject, especially as
regards liquids, will be fully treated later, so that we need
not further consider it here.

54. But let us contrast, with the behaviour of the
particles of a solid or a liquid, that of the particles of a
gas or vapour. Such substances require to be subjected

TERMS APPLIED TO MATTER. 45

to external pressure in order to prevent their particles
from being widely scattered. When a small quantity of
air is allowed to enter an exhausted receiver it dilates so
as to occupy with practical uniformity the whole interior
of the receiver, however large that may be.

This result was, naturally enough, at first ascribed to
a species of repulsion between the various particles ; but
the notion was found to be an erroneous one. For the
effects of a true repulsion, capable of producing the
practically infinite dilatation already spoken of, could not
all be consistent with the corresponding observed results.
The mode of departure from them depends upon the law
according to which the repulsion may be supposed to vary
with the distance between two particles. Some assumed
laws would give as a consequence that the particles would
all be driven to the sides of the vessel, leaving the interior
void. Others would require that the pressure should
change in value if we were to take half the gas and con-
fine it in a vessel of half the content. Others would
make it different at different parts of the surface unless
the vessel were truly spherical, etc. etc.

The true explanation of the phenomenon becomes
obvious to us when we apply heat to the gas. For it
then appears that the pressure requisite to maintain the
whole at a constant volume increases as the temperature
is raised ; and thus that heat is, in some way, the cause
of the pressure.

55. Hence we are led to what is called the Kinetic
Theory of gases, whose fundamental assumption ( 33) is
that the particles dart about in all directions (with an
average speed which is greater the higher the tempera-
ture), impinging on one another, and also upon the sides
of the containing vessel. This continued series of very

46 PROPERTIES OF MATTER.

small but very numerous impacts (each, by itself, absolutely
escaping observation) is perceived by our senses as the
so-called "pressure" exerted by the gas. Experiment
shows that, when a gas is confined in a vessel of definite
size, the changes of its pressure are nearly proportioned
to the changes of temperature, as measured by a mercury
thermometer, whether these changes be in the direction
of a rise or a fall. If we assume, for a moment, that
this statement is true for all ranges of temperature, even
beyond those attainable in experiment, it leads us to the
very important question : At what temperature does the
pressure of a gas vanish ?

Calculations carried out in the above way showed that,
under the assumption just mentioned, all of the (so-called)
permanent gases cease to exert pressure at one common
temperature (about- 273 C.). Therrnodynamical theory
comes to our assistance and shows that the above guess is
not far from the truth : that a body, cooled to 274 C.,
cannot be cooled any farther; that it then is, in fact,
totally deprived of heat

We might, therefore, fancy that a gas, if it could be
brought to this temperature, would be reduced to a mere
layer of incoherent dust or powder, deposited by gravity
on the lower surface of the containing vessel. But
experiment has shown that gaseous particles, even while
in motion, if only close enough together, exert mutual
molecular forces, so that the result (on any gas) of the
conditions above specified would probably be its assuming
a liquid or even a solid form. In fact Andrews has shown
that, for each particular substance, there is a temperature
(usually called the Critical Point) such that, while the
substance is at any higher temperature it is necessarily a
gas, in the sense that it cannot be liquefied by any pressure,

TERMS APPLIED TO MATTER. 47

however great. At any lower temperature it can be
liquefied by the application of sufficient pressure, and is
therefore to be regarded as a true vapour.

56. We speak of bodies as Hard and Soft. These are
barely scientific terms; because, unless they are strictly
defined, they may bear a great variety of meanings.

Thus, for instance, we have the mineralogist's Scale of
Hardness, which is often of great practical value in field-
work. For there are numerous instances in which two
quite different minerals (sometimes a very valuable and a
very common one) are almost undistinguishable from one
another so far as colour, density, and crystalline form are
concerned. Chemical tests (even the comparatively coarse
blowpipe tests), though they would settle a question of
this kind at once, are not readily applied in the field.
Hence the use of the scale of hardness, in which minerals
are so arranged that every one can scratch the surface of
any other which is lower in the scale. By carrying a set
of twelve small specimens only, of selected minerals, the
finder of a doubtful crystal can readily determine its rank
among them as regards scratching; and can thus often
settle in a moment what would otherwise require some
time, even with the facilities of a laboratory.

In such a scale diamond, of course, stands at the top,
while native copper, one of the toughest of substances, is
far below it.

But if we were to test relative hardness by some other
method, say by blows of a hammer, we should be led to
arrange our specimens in a very different order. The scale
above spoken of is, therefore, by no means a scientific one ;
though, as we have seen, it may often give easily some
useful information.

CHAPTER IV.

TIME AND SPACE.

57. WE begin with an extract from Kant, who, as mathe-
matician and physicist, has a claim on the attention of
the physical student of a very different order from that
possessed by the mere metaphysicians.

" Time and space are two sources of knowledge, from
which various a priori synthetical cognitions can be
derived. Of this pure mathematics give a splendid
example in the case of our cognitions of space and its
various relations. As they are both pure forms of
sensuous intuition, they render synthetical propositions a
priori possible. But these sources of knowledge a priori
(being conditions of our sensibility only) fix their own
limits, in that they can refer to objects only in so far as
they are considered as phenomena, but cannot represent
things as they are by themselves. This is the only field
in which they are valid ; beyond it they admit of no
objective application. This peculiar reality of space and
time, however, leaves the truthfulness of our experience
quite untouched, because we are equally sure of it,
whether these things are inherent in things by themselves,
or by necessity in our intuition of them only. Those,
on the contrary, who maintain the absolute reality of

TIME AND SPACE. 49

space and time, whether as subsisting or only as inherent,
must come into conflict with the principles of experience
itself. For if they admit space and time as subsisting
(which is generally the view of mathematical students of
nature), they have to admit two eternal infinite and self-
subsisting nonentities (space and time), which exist with-
out their being anything real only in order to comprehend
all that is real. If they take the second view (held by
some metaphysical students of nature), and look upon
space and time as relations of phenomena, simultaneous
or successive, abstracted from experience, though repre-
sented confusedly in their abstracted form, they are
obliged to deny to mathematical propositions a priori
their validity with regard to real things (for instance in
space), or at all events their apodictic certainty, which
cannot take place a posteriori, while the a priori concep-
tions of space and time are, according to their opinions,
creatures of our imagination only." l

On matters like these it is vain to attempt to dogma-
tise. Every reader must endeavour to use his reason, as
he best can, for the separation of the truth from the
metaphysics in the above characteristic passage.

58. We must now take up, as indicated in 21, the
property Extension, which is one of those expressly in-
cluded in our provisional definition of matter.

It implies that all matter has volume, or bulk. The
thinnest gold leaf has finite thickness, ths finest wire has
a finite cross section.

In popular language this is recognised by the use of
the associated terms length, breadth, arid thickness.

In other words, the term extension recognises the
essentially Tridimensional character of space.

1 Critique of Pure Reason. Max Muller's Translation.

50 PROPERTIES OF MATTER.

Why space should have three dimensions, and not
more nor less, is a question altogether beyond the range
of human reason. Only those who fancy that they know
what space is, would venture (at least after well con-
sidering the meaning of their words) to frame such a
question.

59. The proof that our space has essentially three
dimensions is given in its most conclusive form by the
statement, based entirely upon experience, that

To assign the relative position of two points in space,
three numbers (of which one at least must be a multiple of
the unit of length) are necessary, and are sufficient.

It is an easy matter for us, accustomed to tridimen-
sional space, to imagine one or more of its dimensions
to be suppressed. In fact so-called Plane Geometry is
the geometry of one particular kind of two-dimensional
space; Spherical Trigonometry that of another. We
cannot well speak of the geometry of space of one, or of
no dimensions ; but the idea we should thus attempt to
express is a correct one, though the term is inappropriate.

When, however, we try to conceive space of four or
more dimensions, we are attempting to deal with some-
thing of which we have not had experience ; and thus,
though we may by analogy extend our analytical and
other processes to an imagined space, in which the rela-
tive position of two points depends on more than three
numerical data, we can form no precise idea of how the
additional dimensions would present themselves to our
senses or to our reason.

A few remarks on this subject will be made at the end
of the chapter.

60. Space of no dimensions is a geometrical point, of
which nothing further can be said

TIME AND SPACE. 51

61. Space of one dimension: let us call that dimen-
sion Length : is a mere geometrical line which may be
curved or straight. But to be sure of the existence of
this characteristic, and to understand its true nature, we
must have cognisance of space of two dimensions if it be
a plane curve, of three if it be tortuous. The study of
all the properties of space of one dimension, though an
excessively simple affair, is of very great intrinsic import-
ance, besides being a necessary step towards that of the
higher orders. We will, therefore, treat it so fully that a
far less amount of detail will be necessary when we come
to two and to three dimensions.

62. Every one, whether he be aware of the fact or not,
is acquainted by experience with at least the elements of
this subject. Suppose, for instance, we take as our one-
dimensioned space any one of the roads or railways lead-
ing from Edinburgh to London ; which we will, for the
moment, suppose to be straight, and to run due south.
The mile-stones, set up at equal distances along the road,
mark the positions of various points in terms of the one
dimension, length, which is alone involved, or, rather, to
which for the present we restrict our consideration. And
a Gazetteer or a Railway Guide gives us the positions of
the towns or stations along the road or line : the position
of each being fully described by a single number, under-
stood as a multiplier of a mile or of some other specified
unit of length, and with a qualification which will
presently be introduced.

But these numbers refer to the distance from some
assumed starting-point, or Origin as it is technically
called ; say, in this case, London. Thus we find in an
old Road Guide, for the particular one-dimensioned space

52 PROPERTIES OF MATTER.

called the East Coast Route, a column of data from which

we extract the following :

Miles.
London .........

York .196

Berwick 337

Edinburgh ........ 395

Fractions of miles are omitted, to avoid mere arithmetical
complication.

From this table, by ordinary subtraction, we form a
list, as below, of the lengths of what we may call the

various stages of the route. Thus

Miles.
London to York . , . . . . .196

York to Berwick . 141

Berwick to Edinburgh . . . . .58

It will be seen that, in this list, the origin from which
each number is measured is the first named of the two
corresponding places, and the number itself is found by
subtracting, in the first list, the number corresponding
to the first of the two places from that corresponding to
the second.

63. Now let us at once take the only step which
presents any difficulty. Choose York as our origin, and
boldly apply the rule just given, no matter what the

consequences may be. The result is

Miles.

London . . . -196

York

Berwick. . V 141

Edinburgh 199

Here there is no difficulty whatever in understanding the
numbers for Berwick and for Edinburgh. They are, as
before, the numbers of miles by which Berwick and
Edinburgh are separated from York. Also the number

TIME AND SPACE. 53

for London, when York is the origin, differs from that for
York, when London is the origin, only by change of
sign.

So that we at once recognise the meaning of the
negative sign as applied to a length in our one-dimen-
sional space : it measures the length in the opposite
direction to that in which a positive length is measured.

The necessity for this convention, and its extreme
usefulness, were early recognised in Cartesian geometry,
but they had long before been applied in common arith-
metic as well as in algebra.

Perhaps the simplest view we can take of the subject
is that afforded by a man's " balance " at the bank. So
long as this is on the right side (i.e. positive) he can draw
any less amount and still be on the credit side; if he
overdraws (i.e. takes more out of the bank than his
balance), the difference is negative, and he is to that
amount indebted to the bank.

64. In the first of the three little tables above, all the
places involved lay to the north of the origin (London),
and were all therefore affected by the same sign (which
we happened to take as + ). When York was taken as
origin, Berwick and Edinburgh were to the north, and
their numerical quantities were still + . But London
being to the south, had a - number.

It would be easy to give multiplied examples of this,
but they are unnecessary. The only additional com-
ments which we need make are these :

(1.) When the northern direction along a line was
called +, the southward necessarily became . Simi-
larly had we chosen southward as + , northward would
have become - .

(2.) We chose for our special example a northward-

64 PROPERTIES OF MATTER.

running line, but we might equally well have chosen an
eastward one, etc. Hence pairs such as N". and S., E.
and W., up and down, etc., must he regarded as having
their members contrasted exactly as are the + and - of
Algebra or of Analytical Geometry.

And, just as a displacement in either direction along a
line may be regarded as +, while a displacement in
the opposite direction must then be regarded as - , so it
is with rates of motion, i.e. Speeds, in space of one
dimension.

Thus the relative speed of two trains running north-
ward, A at 60 miles an hour, B at 40, is 20 miles an
hour northward as regards A seen from B, and 20 south-
ward as regards B seen from A; so if A be moving
southward at 60 miles an hour, and B northward at
40, the speed of A with regard to B is 100 miles per
hour southward, and of B with regard to A 100 miles per
hour northward.

The idea of speed, as so many units of linear space
described per unit of time, is a complex one : involving
both of the fundamental ideas. We express this by
saying that its Dimensions are

L

[*}

This implies that, in whatever proportion we increase
our unit of length, the measure of a speed is diminished
in that proportion : while it is increased in the same
proportion as that in which the unit of time is
increased.

Thus a speed of 5280 feet per second is but 1 mile
per second; while a speed of 1 foot per second is 60
feet per minute.

65. A precisely similar distinction (as to + and - ) is

TIME AND SPACE. 65

observed when our one-dimensional space is a curved
line; take for example the orbit of a planet. To
describe fully the position of the planet, when the orbit
is given, one number alone (say the angle-vector, the
angle which the radius-vector, or line joining the centres
of planet and sun, makes with some fixed line in the
plane of the orbit) is required. This, however, must
again be qualified as + or - . (In the case of angles, we
agree to call them + when they are measured in the
opposite direction to that of the motion of the hands of a
watch; that is, when they are described in the same
sense as that in which the northern regions of the earth
turn about the polar axis.) Angular velocity in one plane
(i.e. rate at which the radius-vector turns) is similarly
characterised.

In aU cases where motion is restricted to one line the
same thing holds. Thus the position of a pendulum is
at every instant completely assigned by the angle the
rod makes with the vertical, provided we are also told on
which side the displacement is.

The record kept by a self-acting tide-gauge gives at
any instant the elevation or depression (again + and )
of the water above or below the mean level. Similarly
with registering barometers, thermometers, etc. But,
for the full appreciation of the indications of these
records, they are usually made in two dimensions by the
use of an important principle which will presently be
explained. (68.)

66. In what precedes we have been dealing with a
kind of space in which the only displacements are
forward or backward ; nothing is possible (nor even con-
ceivable) sideways or upwards.

This characteristic applies to Time, as well as to space

56 PROPERTIES OF MATTER.

of one dimension, and therefore we should expect to find,
as we do find, that (with the necessary change of a word
or two) all that has just been said with reference to relative
position is true of events in time, as well as of points in
one-dimensional space. There is no such thing as motion
or displacement in time, so that this part of the analogy
is wanting. Every event has its definite epoch, for ever
unalterable. And of course there is no going sideways or up-
wards, as it were, out of the one-dimensional course of time.

Thus we find that to assign definitely the position of
an event in time, provided our origin is assigned, all
we need know is a single number (a multiplier of the
time-unit) with its sign, + or - , signifying time after or
time before that origin.

Our usually adopted origin is the Christian era, and
we speak of 1899 A. D. as the present year [I leave these
dates as written. W. P.], while the date of the battle of
Marathon is recorded as 490 B.C. The difference between
the characteristics A.D. and B.C. is of precisely the same
nature as that between north and south, or + and .

Hence, if we wish to find the interval between the
present time and the battle of Marathon, we have to
subtract +1899 (the position of the new origin) from
- 490. The result is - 2389, i.e. Marathon was fought
2389 years ago. Thus to change the origin, or epoch,
we must perform precisely the same operation as that
which gave us the table in 63, from the first table in
62. Similarly, to change our system of chronology to
the year of the world (designated by A.M.) or to the old
Roman (marked A.U.C.), all we need do is to subtract from
each date (A.D. or B.C., regarded as + and - respectively)
the assumed date of the creation of the world (4004 B.C.)
or of the foundation of the city of Rome (753 B.C.).

TIME AND SPACE. 57

We need say no more on such a matter. Every intel-
ligent reader can make new and varied examples for
himself.

67. Passing next to space of two dimensions, whether
plane or spherical, we see at once from a map, or a globe,
that the position of a place is given by two numbers, its
Latitude and Longitude. But each of these has to be
qualified for definiteness by the + or - sign, or something
equivalent. Thus we have K or S. latitude, and E. or
W. longitude.

But there are two methods, specially applicable to the
plane, which deserve closer attention in view not only of
their intrinsic usefulness, but also of their bearing on the
general question of tridimensional space. These are known
in geometry as Rectangular and Polar co-ordinates.

68. In the first we assume two reference lines at right
angles to one another, both passing through the origin,
and assign the position of a point by giving its distances
from these two lines. These distances are looked on as
drawn towards the point from either line, and each there-
fore changes sign when the point is taken on the other
side of the corresponding reference line. This is symbol-
ised in the cut. Ox, Oy are the two reference lines, the
origin. The perpendiculars PM, PN, let fall from P on
these lines, completely, and without ambiguity, define its
position. For if we know OM or NP, the # of P, i.e. its
distance from Oy, that condition alone limits our choice
for P to points lying in PM, a line drawn parallel to
Oy and everywhere at the assigned distance, x, from it.
Similarly, y being given as ON or MP, the choice
of points is limited to those on the line NP, all of which
have this property.

But two lines at right angles to each other must

58

PROPERTIES OF MATTER.

intersect, and in one point only. Thus the point P is
determined by the conditions without ambiguity.

-X

X

FIG, 2.

If P lie to the left of Oy, its x is negative ; if below
Ox, its y is negative. The lettering in the cut, at the ends
of the lines bounding the four quadrants, shows at a glance
the signs of x and y when P is situated in any one of them.

In general, any given relation, between the x and y of
a point, limits its position to a definite Curve in the plane
of the reference lines. It is often very convenient to
represent such a relation by a curve ; and, in fact, most
self-registering instruments actually trace such a curve
for us. Thus, if intervals of time (as OM) measured
from a definite instant (represented by 0) be laid off
along Ox, with the corresponding heights of the thermo-
meter, barometer, tide, etc., erected as perpendiculars
(MP) at their extremities, we have (as the Locus of P) a
curve showing the mode in which temperature, pressure
of the atmosphere, etc., change as time goes on. But
such curves can be traced by a pencil attached to the
instrument, or by photographic processes, on a long band

TIME AND SPACE. 50

of paper which is drawn horizontally past it, at a uniform
rate, by clockwork.

69. In the second method mentioned in 67 the data
are the length of OP (the radius-vector), and the magni-
tude of < MOP (the angle-vector), 65. These are usually
denoted by r and 9, respectively. Here r is always taken
as a positive (or rather a signless) quantity, while 6 is posi-
tive if it be measured round from Ox counter-clockwise.

This is the method adopted by a surveyor when, with a
chain and a theodolite, he measures a field. His reference
line, Ox, is usually given by a magnetic needle attached to
the theodolite. He measures the angle a;OP and the dis-
tance OP, P being a corner (let us say) of the field. These
two data, determined for each prominent part of the bound-
ary, enable him to plot the field ; and therefore contain all
the necessary numerical data for calculating its area, etc.

It is also the method usually employed in dealing with
orbital motion of any kind in one plane.

Comparing the two methods, we see that the directed
line OP may be resolved (as it is called) into OM and
MP, lines in directions perpendicular to one another.
Also that this resolution, in any direction, is effected by
means of the cosine of the angle involved.

<
For x = OM = OP cos zOP = r cos 6,

<
y= MP = OP cos ?/OP = r sin 6.

It is clear that, though we have hitherto spoken of
and P as the simultaneous positions of two points, we
may look on them as successive positions of one (moving)
point. If we look on the displacement as having been
produced uniformly, and in one second, it represents in
magnitude and direction the Velocity of the moving

60 PROPERTIES OF MATTER.

point ; and OM, MP represent, on the same scale, its
resolved parts or components, parallel to Ox and O?/.

These components are entirely independent of one
another, so that to compound two or more displacements
or velocities we have only to resolve each into its east-
ward and its northward components, and deal with these
respectively by the ordinary algebraic process, to obtain
the corresponding components of the resultant.

70. As examples, we give one or two results which
will be specially useful to us in later chapters.

If a point be moving, in any manner whatever, we may
consider its velocity alone, independent altogether of the
actual path pursued. Here we are introduced to a new
idea, that of Acceleration. For, as velocity is rate of
change of position, acceleration is rate of change of
velocity.

Take any fixed point, 0, and let OP represent, in
magnitude and direction, the velocity of the moving

point. After one unit of
time let the velocity be
represented by OP X ; after
two units, by OP 2 ; and
so on. It is clear that all
the points P, P v P 2 , etc.,
lie on some definite curve,
which will be the more

accurately traced the greater the number of points we
obtain in any assigned portion of it ; i.e. the smaller we
assume our unit of time. If the motion whose properties
are thus studied be that of a particle of matter, this
curve (which is called the Hodograpli) is necessarily
continuous, for the velocity cannot alter by starts ( 120)
either in magnitude or in direction. And, as OP passes

TIME AND SPACE. 61

to a proximate position, OQ, by having a velocity PQ
compounded with it, the Acceleration of OP is the
velocity with which P moves in its curve. If the path
be a plane curve, the hodograph is also plane.

This construction enables us at once to solve a number
of elementary problems in kinematics, which will be of
great use to us in the sequel.

In 64 above, we showed that the dimensions of speed
(V) are

In precisely the same way we see that those of accelera-
tion (A) are

Thus the numerical measure of acceleration is diminished
in proportion as the unit of linear space is increased :
but is increased in the duplicate ratio of that of the time
unit.

An acceleration of 1 foot per second, per second, is
obviously the same as 3600 feet per minute, per minute.

71. Suppose a point to move uniformly, with speed V,
in a circle of radius R. OP in the hodograph (Fig. 3)
has constant length V, and its direction rotates uniformly.
Hence the hodograph is another circle, also uniformly
described in the same sense (i.e. clockwise or counter-
clockwise), and in the same period of time. Hence the
speed of P must be such that it describes a circle of
radius V, in the time that a point whose speed is V takes
to go round a circle of radius R. It must, therefore, be
V 2 /R. Also the direction of this speed is perpendicular
to OP, and therefore along the radius of the first circle.
And its direction is towards the centre of that circle,

62 PROPERTIES OF MATTER.

because both circles are described clockwise, or counter-

clockwise.

Let, now, the figure repre-
sent the circle of radius R,
and draw any diameter, ACA'.
Then N moves round with
speed V, and the acceleration of
its motion is V 2 /R along NC.
Remembering that accelerations
and velocities are resolved like
lines, we see that if NM be

drawn perpendicular to AA', the speed of the point M

along MC will be

V MN
NO'
and its acceleration along MC, and towards C, will be

R CN = JR 2 ^ M *

The motion of M, thus defined, is called Simple Har-
monic. It obviously consists in a vibration back and
forward along the line AA', the speed being greatest at
C, and vanishing at A and A'. The special characteristic
is that the acceleration is always directed towards C, and
is proportional to the displacement of M from that point.

72. If we use Newton's Fluxional Notation, in which
the rate at which a quantity increases per unit of time
is expressed by putting a dot over the symbol for that
quantity, a second dot placed over it will signify the rate
at which that rate increases, and so on.

Thus, if CM above be denoted by x, the speed of M is
x, and its acceleration is x. And we see ab once from the
result of last section that

TIME AND SPACE. 68

the negative sign being prefixed because while x is directed
to the right in the figure, the acceleration is directed to
the left, and conversely. Whenever, in future chapters,
we meet with a relation of this kind, we will, therefore,
interpret it as expressing simple harmonic motion. The
multiplier of the right-hand side depends only on the ratio
of V to E : what is called ( 65) the angular velocity of
the radius-vector CN. If we denote this by w, the equa-
tion may be written

X = -u?x;

an equation which belongs to all simple harmonic motions,
whatever be their range of vibration, provided the angular
velocity in the corresponding unform circular motion be
(o, or the period of a complete revolution 27r/o>. Any such
motion is therefore fully described by

05 = a COS.

where a and a are absolutely arbitrary.

73. The result above was obtained by projecting uni-
form circular motion on a diameter of the circle, or, what
comes to the same thing, on a plane perpendicular to the
plane of the circle.

But an exceedingly interesting result is obtained by
projecting the circular motion on any other plane. In
orthogonal projection equal areas are projected into equal
areas, and a circle is projected into an ellipse whose centre
is the projection of the centre of the circle.

Hence the projection gives motion in an ellipse, the
radius-vector drawn from the centre of the ellipse tracing
out equal areas in equal times, and the acceleration being
still directed inwards along the radius-vector, and still
bearing the same proportion to it.

74. Another extremely useful result may be obtained

64

PROPERTIES OF MATTER.

by supposing the unform angular velocity in the circle
to be maintained, but with a continual shrinking of the
radius at a rate measured (per second) by K times its
length at each instant.

The velocity of the moving point is thus made up of
two components, one along the circle, the other along the
radius, each proportional to the radius. Hence the path
is a spiral which makes a constant angle with the radius,
what is called the Equiangular, or Logarithmic, Spiral.
The radius- vector still revolves uniformly. 1

Let PB be the spiral, SP any radius. Then, if PT be

FIG. 5.

the velocity of P, and a the (constant) angle between its
direction and that of PS, we see at once that

whence

PT sin a = o>SP, PT cos a = /cSP,
K = <a cot a.

If SO be equal and parallel to PT, Q is a point in the
hodograph. But as PT, and therefore SQ, is proportional
to SP, and the angle QSP is the supplement of a, the
hodograph is the same spiral rotated through a given
angle, and altered in its linear dimensions by the factor

Thus the holograph of the hodograph is another

l Proc. K.S.E., December 19, 1867.

riME AND SPACE. 65

similar spiral, again turned through the same angle, and

with its radii altered in the ratio ( -^ ) . If PU be

\sm a/

drawn, making an angle a with TP produced backwards,
and meeting QS in U, it will therefore be the direction of
acceleration at P.

But PU may be resolved into PS, SU, the first of
which is along the radius-vector, the second parallel to
the tangent at P. The parts of the acceleration in these
directions are, respectively,

The latter of these, by the first equation above, may be
written as

2(^-yPT slnaCOSa ~2c, cota.PT-2PT.
\sm a/ <

Hence the motion of P is due to an acceleration along,
and proportional to the length of, PS, and another along,
and proportional to the length of, TP.

And of course the resolved part of the motion along
any line in the plane possesses the same characteristics.
If x represent the distance between the projections of S
and of P, on such a line, we see at once that we have

r- 2
x= -2ocoto. x-

-S-
Sin 2 a

or, introducing the value of K above,

This differs from the equation for simple harmonic
motion ( 72) by the term involving K, whose physical
interpretation (when it is multiplied by the mass of the
oscillating particle) is a resistance proportional to the
speed of the motion. But the preceding investigation

E

66 PROPERTIES OF MATTER.

shows us that an equation of this form represents the
resolved part (in some definite direction) of uniform
circular motion with angular velocity to, the radius of the
circle shrinking in each second by the fraction K of its
amount. (This is the same thing as saying that its
logarithm diminishes by K in unit of time.) Or we may
call it simple harmonic motion whose scale is constantly
diminishing at a definite rate.

This special case of motion is fully described by the
equation

Compare the result of 72.

75. The step to three-dimensional space is now easy.
We will take it from a somewhat altered point of view.
Our reference system is now three planes at right angles
to one another ; say the floor, the north wall, and the
west wall of a room, the corner in which these three meet
being for the time our origin.

And the position of a point is determined without
ambiguity if we know its distances from these planes,
with the proper sign of each.

For, knowing only its distance from the floor, we limit
it to the horizontal plane which is everywhere at that
distance from the floor. Similarly the second condition
limits it to a definite plane parallel to the north wall.
These two conditions together limit it to a certain hori-
zontal line lying east and west. The third condition
limits it to a certain plane parallel to the west wall ; and
this is intersected in one point, and one only, by the east
and west line just mentioned. That one is the sole point
which satisfies all three conditions.

Thus, let represent the origin, yOz the north wall,

TIME AND SPACE.

67

zOx the west wall, and xOy the floor. The figure is
drawn as seen by an eye equidistant from these three
planes, and in the room, i.e. on the positive side of each
of them. And it will be noticed that the lettering, x, y, z
of the ends of the edges, which meet in 0, is so applied
that rotation from Ox to Oy, from Oy to Oz, and from
Oz to Ox again, will all alike appear to be counterclock-
wise.

The position of any point, P, is then found thus :
Draw PIS' perpendicular to the floor, meeting it in N ;
thence NM perpendicular to Ox. Then OM = ic is the
distance of P from the north wall ; MN = y is its distance
from the west wall ; and JSTP = z is its distance from the
floor.

If P assume a new position which requires it to pass
through any one of these planes, the corresponding co-

68 PKOPERTIES OF MATTER.

ordinate changes sign ; if it pass through an edge (i.e.
the intersection of two of these planes) two co-ordinates
change sign ; and if it pass through (where the three
planes meet) all three co-ordinates become negative.
This is illustrated by the negative lettering at the (dotted)
prolongation of each edge through 0.

76. But, in analogy with the second method of 69,
we see that the position of P will be fully specified if
we know the vertical plane through in which it lies
(i.e. the plane zON), the angle NOP in that plane, and
the length of OP. The first is determined if we know
the angle xON. Hence we determine P by its distance
from 0, and two angles which (together) enable us to
assign the direction of OP. The angle icON is called the
Azimuth of the plane zQN ; let us denote it by 0. The
angle NOP is called the Altitude of P, as seen from ;
let us denote it by <. Also let the length of OP be, as
before, called r.

Comparing, as before, the results of the two methods,
we see that ON = r cos <f>, and therefore

x = OM = ON cos 6 = r cos Q cos 6,
y = MN = ON sin = r cos <p sin 0,
z = NP =rsinp.

The elements of spherical trigonometry show that the
multipliers of r, in the values of x, y, z respectively, are
the cosines of the angles between the line OP and the
lines Ox, Oy, Oz. Hence the more symmetrical method,
in which these cosines are represented by I, m, n respec-
tively, gives

x = rl, y^rm, z = rn,

with the condition

TIME AND SPACE. 69

It is easy to see that the remark in 69, as to resolu-
tion of a velocity in two dimensions, holds with respect
to three.

Then Newton's Second Law of Motion (Chap. VI.) at
once extends these conclusions to Forces.

77. A remark of great importance must be made here.
We saw in 68 that a point was determined, in x, y co-
ordinates (i.e. plane space of two dimensions), as the
intersection of two straight lines, to one of which it was
confined by its x being given in value, to the other by
the value of its y. But any two independent conditions
connecting x and y will, also, determine their values. A
single condition connecting x and y is known as the
Equation of a Curve, and, when given, limits the position
of P to that curve. Two such conditions, therefore, give
P by the intersection of two curves, on each of which it
must lie. Such a condition applied to a physical particle
is called a Degree of Constraint. In two-dimensional
space a free particle has but two Degrees of Freedom, one
of which is removed by each degree of constraint to which
it is subjected.

78. Similarly we saw that, in three dimensions, the
point given by x, y, z is determined as the intersection of
three planes, on each of which it must lie. But any one
condition connecting the values of x, y, and z is the
Equation of a Surface, and, when it is given, a particle
at the point is subjected to one degree of constraint.
When free, it has but three degrees of freedom ; and thus
three degrees of constraint, by completely determining its
x, y, and z, fix its position.

We should arrive at the same result by considering
relations among the r, 0, <f> co-ordinates. But it suffices
to consider merely what species of constraint each of

70 PROPERTIES OF MATTER.

these imposes when its value is given. All points for
which r has a given value lie on a sphere whose centre is
at 0. When is given, the point must lie somewhere
in the vertical plane zOK When < is given, it must lie
somewhere on a right cone of which is the vertex and
Oz the axis.

[The two latter statements are easily illustrated by
means of a telescope, mounted (in the common way) on
a stand which allows it to rotate either about a horizontal,
or about a vertical, axis. Place it in any azimuth, and
vary its altitude, it turns in a vertical plane about the
horizontal axis. Place it at any altitude, and vary its
azimuth, it rotates conically about the vertical axis.
Hence, by means of these co-ordinates, or conditions,
each definite point in its axis is constrained to lie on a
sphere, a plane, and a cone, simultaneously.]

79. Two devices are in common use for enabling us to
represent, on a plane (or other space of two dimensions)
the third dimension.

Thus, in an Admiralty chart, we find the sea-area
marked over with figures denoting Soundings : i.e. the
average depth of the water at certain places is written in
in fathoms. These soundings are of course more numerous
in regions where there are shoals or intricate channels.
But it is obvious that, if they were numerous enough,
they would enable us to construct a model of the sea-
bottom. The soundings, therefore, supply, as it were,
the necessary third dimension. But this process, though
usually sufficient for purposes of navigation, is at best a
rude and incomplete one.

The other method, however, rises to a very high order
of scientific importance, not merely from the point of view
for which it was originally devised, but on account of the

TIME AND SPACE. 71

extent to which its essential principles are now applied
throughout the whole range of physics. We therefore
devote some space to its full explanation.

80. This is called the method of Contour Lines, and is
employed with great effect in the best maps, such as those
of the Ordnance Survey.

A contour line passes through all places which are at the
same height above the sea-level.

Thus the sea-margin is itself the contour line of no
elevation. Suppose the water to rise one foot (vertically).
There would be a new sea-margin, in general encroaching
more on the land than the former ; encroaching most at
places where the beach has the gentlest slope, not encroach-
ing at all on a perpendicular cliff, and thrust out (seawards)
from an overhanging cliff. This is the contour line of one
foot elevation. It is clear that by supposing a gradual
rise of the sea, or subsidence of the land, foot by foot,
we could obtain a series of curves (each in its turn a
sea-margin) gradually circumscribing the uncovered portion
of the land, and finally closing in over its highest peak.
We require no such natural convulsion as that just im-
agined. Cloud strata, or fog-banks, with definite horizontal
surfaces, constantly show us these appearances in hilly
countries. But it is a simple matter of Levelling to trace
out contour lines, and to draw them on a map of the
district. For practical purposes it is usually sufficient to
draw them for every 50 or 100 feet of additional elevation
above the sea-level.

The celebrated Parallel Roads of Glen Roy are merely
contour lines, etched on the sides of the valley by long-
continued but slight agitation of the margin of the water
which filled the glen to various depths in succession, as the
barriers which dammed it ut> were, at intervals, broken down.

72

PROPERTIES OF MATTER.

Eeferring to 78, we see that a surface can be fully
described in terms of one relation among x, y, and z. Let
the plane of Oar, Oy, be that of the sea-level, and let the
relation expressing the surface of the land be

f(x, y, s) = 0.

Then the contour lines, as traced on the (two-dimension)
map are the curves

f(x, y, 0)-0,
f(x t y, 100) = 0,
f(x, y, 200) = 0, etc.

81. To familiarise the student with the general appear-

FIG. 7.

ance of contour lines, and their relation to the form of the
corresponding surface, we give those of a right cone whos^

TIME AND SPACE. 73

axis is vertical, of a hemisphere, and of a fusiform or
spindle-shaped body.

The fusiform body, whose contour lines are drawn, is
formed by the rotation of a quadrant of a circle about a
vertical tangent, the point of contact being the apex.
And the contour lines are drawn, in each case, at suc-
cessive heights increasing by one-fifth of the whole height
of the figure. Thus the distances between successive
contours, in the two last figures, form the same series of
values, but in opposite order.

The equality of distance between the successive contour
lines of the cone indicates uniform steepness throughout.
In the hemisphere the lines are closer together near the
boundary of the figure, in the spindle they close in on one
another towards the centre ; the hemisphere being steepest
at its edges, and the spindle surface steeper towards the
point.

82. In fact, the Gradient of a surface in any direction
(i.e. the amount of rise per horizontal foot) is obviously,
at any point, inversely as the distance in that direction
between successive contour lines, for they are traced at
successive equal differences of level ; and thus the dis-
tance between them, along any line drawn on the map,
is the space by which we must advance horizontally
along that line while ascending or descending vertically
through 100 feet.

83. The line of steepest slope at any point of a surface
is, of course, perpendicular to each contour line where it
meets it. For the contour line is horizontal, i.e. has nc
slope. And in the projection on a horizontal plane this
perpendicularity remains. Thus the line of greatest slope
at any point is represented on the map by the shortest
line which can be drawn from that point to the nearest

74

PROPERTIES OF MATTER.

contour line. It is the path along which a drop of water
would trickle down. It is therefore called a Stream-line.
84. If the surface be like that of a saddle (concave up-
wards along the horse's back, convex upwards across it),
we have at the middle of the saddle what is called, in
geography, a Col or mountain-pass : the lowest point
of the ridge between two neighbouring summits. The
characteristic of the col is that, at such a point, a contour
line intersects itself. The following sketch shows the
general form of the contours near such a point.

FIG. 8.

In the shaded regions depicted to the right and left oi
the col the ground rises, in the unshaded regions depicted
above and below it falls. [The figures on the contour
lines show their order of altitude above the sea-level.]

Other very special peculiarities might be mentioned,
but they are not necessary for the beginner; and the
more mathematical reader can easily work them out for
himself. 1

iSee Cayley, Phil. Mag., XVIII. 264 (1859); Clerk-Maxwell,
Ibid., December 1870.

TIME AND SPACE. 76

85. If we draw, by the help of the contour lines, the
stream-lines (which, 83, cut them at right angles), we
find that they have the following property. In regions
above the level of a col, they fall away on both sides from
that particular one of their number which passes from a
mountain Summit down to the col, and thence up to the
neighbouring summ